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<c val="#800000">Using Units</c>
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<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">One of the useful features of Mathcad is its ability to track standard units during calculations and to convert the units of calculated quantities automatically. You can mix and match units as you like and be assured of getting the correct result. This worksheet lets you practice using built-in units and producing results in the units you want. </p>
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<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">By default, Mathcad uses units from the SI unit system (International System of Units). Mathcad offers several choices of unit systems: SI, CGS, MKS, U.S. customary units, or no unit system. </p>
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<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Here is a calculation that mixes units:</p>
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<ml:real>5</ml:real>
<ml:id xml:space="preserve">ft</ml:id>
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<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Although each term in the calculation above is expressed in a different unit of length - feet (ft), inches (in), meters (m), and yards (yd) - the result is appropriately scaled and given in the base SI unit for length, meters. </p>
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<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Here is another example in which units are mixed:</p>
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<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">The two terms in this expression have mismatched units: one is a unit of length (ft) and one is a unit of time (sec). Mathcad knows better than to try to come up with a result. Click on the expression to see the error message - it tells you that the second argument must have a dimension of Length in order to be added to the first.</p>
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<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To Enter Expressions Containing Units</p>
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<phrase>units in expressions</phrase>
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</ml:apply>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:real>10972</ml:real>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="second"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
</math>
<rendering item-idref="5"></rendering>
</region>
<region region-id="1510" left="6" top="618.75" width="27.75" height="14.25" align-x="6" align-y="630" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Try it!</p>
</text>
</region>
<region region-id="1916" left="186" top="618.75" width="133.5" height="57" align-x="192.75" align-y="630" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<b>
<c val="#008080">Tip</c>
</b>: This result appears in seconds (s), the default base unit. Seconds can be abbreviated as "s" and "sec."</p>
</text>
</region>
<region region-id="1505" left="6" top="698.25" width="24.75" height="12" align-x="17.25" align-y="708" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Type:</p>
</text>
</region>
<region region-id="1580" left="186" top="698.25" width="20.25" height="12" align-x="193.5" align-y="708" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
</text>
</region>
<region region-id="1527" left="6" top="729" width="135.75" height="10.5" align-x="11.25" align-y="738" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="typing" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">55mi/hr [Spacebar] *35sec =</p>
</text>
</region>
<region region-id="1581" left="186" top="718.5" width="132" height="33.75" align-x="251.25" align-y="738" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:real>55</ml:real>
<ml:apply>
<ml:div></ml:div>
<ml:id xml:space="preserve">mi</ml:id>
<ml:id xml:space="preserve">hr</ml:id>
</ml:apply>
</ml:apply>
<ml:real>35</ml:real>
</ml:apply>
<ml:id xml:space="preserve">sec</ml:id>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:real>860.55200000000025</ml:real>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="meter"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
</math>
<rendering item-idref="6"></rendering>
</region>
<region region-id="1529" left="6" top="756.75" width="27.75" height="14.25" align-x="6" align-y="768" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Try it!</p>
</text>
</region>
<region region-id="1512" left="6" top="836.25" width="24.75" height="12" align-x="17.25" align-y="846" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Type:</p>
</text>
</region>
<region region-id="1997" left="186" top="836.25" width="20.25" height="12" align-x="193.5" align-y="846" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
</text>
</region>
<region region-id="1515" left="6" top="867" width="29.25" height="10.5" align-x="12" align-y="876" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="typing" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">ft/m=</p>
</text>
</region>
<region region-id="1998" left="186" top="856.5" width="61.5" height="33.75" align-x="204.75" align-y="876" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:div></ml:div>
<ml:id xml:space="preserve">ft</ml:id>
<ml:id xml:space="preserve">m</ml:id>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:real>0.3048</ml:real>
</result>
</ml:eval>
</math>
<rendering item-idref="7"></rendering>
</region>
<region region-id="2000" left="186" top="894.75" width="153.75" height="71.25" align-x="192.75" align-y="906" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<b>
<c val="#008080">Tip</c>
</b>: This result gives you the conversion factor from feet to meters. In other words, multiply a length in feet by this result to get a length in meters.</p>
</text>
</region>
<region region-id="1532" left="6" top="906.75" width="27.75" height="14.25" align-x="6" align-y="918" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Try it!</p>
</text>
</region>
<region region-id="1856" left="0" top="966" width="6000" height="6" align-x="0" align-y="966" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="1542" left="6" top="986.25" width="186" height="12" align-x="24" align-y="996" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To Use the Insert Unit Dialog Box</p>
</text>
</region>
<region region-id="1550" left="6" top="1010.25" width="124.5" height="24" align-x="19.5" align-y="1020" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Select <b>Unit</b> from the <b>Insert</b> menu.</p>
</text>
</region>
<region region-id="2059" left="186" top="1010.25" width="20.25" height="12" align-x="193.5" align-y="1020" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
</text>
</region>
<region region-id="2063" left="162" top="1032" width="253.5" height="207" align-x="162" align-y="1032" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<picture>
<png item-idref="8" display-width="252" display-height="205.5"></png>
</picture>
<rendering item-idref="9"></rendering>
</region>
<region region-id="1556" left="6" top="1046.25" width="142.5" height="36" align-x="18" align-y="1056" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Keep this dialog box open while you perform the next several exercises.</p>
</text>
</region>
<region region-id="2065" left="6" top="1262.25" width="42.75" height="12" align-x="17.25" align-y="1272" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Type:</p>
</text>
</region>
<region region-id="2064" left="186" top="1262.25" width="36" height="12" align-x="193.5" align-y="1272" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
</text>
</region>
<region region-id="1561" left="6" top="1281" width="69.75" height="10.5" align-x="8.25" align-y="1290" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="typing" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">30000kg+1.2*</p>
</text>
</region>
<region region-id="2008" left="186" top="1278.75" width="251.25" height="15.75" align-x="348.75" align-y="1290" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:minus></ml:minus>
<ml:apply>
<ml:plus></ml:plus>
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:real>30000</ml:real>
<ml:id xml:space="preserve">kg</ml:id>
</ml:apply>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:real>1.2</ml:real>
<ml:id xml:space="preserve">tonne</ml:id>
</ml:apply>
</ml:apply>
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:real>120000</ml:real>
<ml:id xml:space="preserve">lb</ml:id>
</ml:apply>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:real>-23231.0844</ml:real>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="kilogram"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
</math>
<rendering item-idref="10"></rendering>
</region>
<region region-id="1565" left="6" top="1304.25" width="167.25" height="60" align-x="16.5" align-y="1314" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Notice that the Insert Unit dialog has scrolled down to Mass and Kilogram. In the Unit list that comes up, click Metric ton (tonne) and then click "Insert." Then type:</p>
</text>
</region>
<region region-id="1573" left="6" top="1377" width="57.75" height="10.5" align-x="6" align-y="1386" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="typing" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">-120000lb=</p>
</text>
</region>
<region region-id="1574" left="6" top="1410.75" width="27.75" height="14.25" align-x="6" align-y="1422" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Try it!</p>
</text>
</region>
<region region-id="1857" left="0" top="1470" width="6000" height="6" align-x="0" align-y="1470" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="1584" left="6" top="1478.25" width="200.25" height="12" align-x="6" align-y="1488" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To Change the Units of a Result</p>
</text>
<indexes>
<phrase>units of a result, changing</phrase>
<phrase>changing units</phrase>
</indexes>
</region>
<region region-id="1603" left="6" top="1502.25" width="42.75" height="12" align-x="17.25" align-y="1512" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Type:</p>
</text>
</region>
<region region-id="2017" left="240" top="1502.25" width="36" height="12" align-x="247.5" align-y="1512" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
</text>
</region>
<region region-id="1606" left="6" top="1527" width="85.5" height="10.5" align-x="8.25" align-y="1536" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="typing" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">3hr+4min-68sec=</p>
</text>
</region>
<region region-id="2018" left="240" top="1524.75" width="151.5" height="15.75" align-x="337.5" align-y="1536" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:minus></ml:minus>
<ml:apply>
<ml:plus></ml:plus>
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:real>3</ml:real>
<ml:id xml:space="preserve">hr</ml:id>
</ml:apply>
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:real>4</ml:real>
<ml:id xml:space="preserve">min</ml:id>
</ml:apply>
</ml:apply>
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:real>68</ml:real>
<ml:id xml:space="preserve">sec</ml:id>
</ml:apply>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:real>10972</ml:real>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="second"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
</math>
<rendering item-idref="11"></rendering>
</region>
<region region-id="1635" left="6" top="1550.25" width="213.75" height="48" align-x="16.5" align-y="1560" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To get a result in minutes, click the result and then click the small <b>black rectangle</b> (it may appear green) to the right of the displayed units. In the <b>units placeholder</b>, type:</p>
</text>
<indexes>
<phrase>units placeholder</phrase>
</indexes>
</region>
<region region-id="2019" left="240" top="1586.25" width="36" height="12" align-x="247.5" align-y="1596" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
</text>
</region>
<region region-id="1627" left="6" top="1611" width="51.75" height="10.5" align-x="13.5" align-y="1620" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="typing" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">min [Enter]</p>
</text>
</region>
<region region-id="2020" left="240" top="1608.75" width="165" height="15.75" align-x="337.5" align-y="1620" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:minus></ml:minus>
<ml:apply>
<ml:plus></ml:plus>
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:real>3</ml:real>
<ml:id xml:space="preserve">hr</ml:id>
</ml:apply>
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:real>4</ml:real>
<ml:id xml:space="preserve">min</ml:id>
</ml:apply>
</ml:apply>
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:real>68</ml:real>
<ml:id xml:space="preserve">sec</ml:id>
</ml:apply>
</ml:apply>
<ml:unitOverride>
<ml:id xml:space="preserve">min</ml:id>
</ml:unitOverride>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:real>182.86666666666667</ml:real>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="minute"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
</math>
<rendering item-idref="12"></rendering>
</region>
<region region-id="1608" left="6" top="1638.75" width="27.75" height="14.25" align-x="6" align-y="1650" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Try it!</p>
</text>
</region>
<region region-id="1643" left="6" top="1706.25" width="215.25" height="24" align-x="17.25" align-y="1716" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Click the result below, then click the <inlineAttr font-weight="normal">units placeholder</inlineAttr> and type:</p>
</text>
</region>
<region region-id="2038" left="6" top="1743" width="45.75" height="10.5" align-x="10.5" align-y="1752" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="typing" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">mi [Enter]</p>
</text>
</region>
<region region-id="2040" left="240" top="1742.25" width="36" height="12" align-x="247.5" align-y="1752" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
</text>
</region>
<region region-id="2031" left="18" top="1762.5" width="135.75" height="33.75" align-x="87" align-y="1782" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:real>55</ml:real>
<ml:apply>
<ml:div></ml:div>
<ml:id xml:space="preserve">km</ml:id>
<ml:id xml:space="preserve">hr</ml:id>
</ml:apply>
</ml:apply>
<ml:real>35</ml:real>
</ml:apply>
<ml:id xml:space="preserve">sec</ml:id>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:real>534.72222222222229</ml:real>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="meter"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
</math>
<rendering item-idref="13"></rendering>
</region>
<region region-id="2032" left="240" top="1762.5" width="119.25" height="33.75" align-x="309" align-y="1782" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:real>55</ml:real>
<ml:apply>
<ml:div></ml:div>
<ml:id xml:space="preserve">km</ml:id>
<ml:id xml:space="preserve">hr</ml:id>
</ml:apply>
</ml:apply>
<ml:real>35</ml:real>
</ml:apply>
<ml:id xml:space="preserve">sec</ml:id>
</ml:apply>
<ml:unitOverride>
<ml:id xml:space="preserve">mi</ml:id>
</ml:unitOverride>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:real>0.33226098473801885</ml:real>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="mile"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
</math>
<rendering item-idref="14"></rendering>
</region>
<region region-id="1680" left="6" top="1844.25" width="215.25" height="24" align-x="17.25" align-y="1854" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Click the result below, then click the <inlineAttr font-weight="normal">units placeholder</inlineAttr> and type:</p>
</text>
</region>
<region region-id="1683" left="6" top="1881" width="63.75" height="10.5" align-x="10.5" align-y="1890" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="typing" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">km/hr [Enter]</p>
</text>
</region>
<region region-id="1690" left="84" top="1880.25" width="9.75" height="12" align-x="88.5" align-y="1890" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">or<sp count="2"></sp>
</p>
</text>
</region>
<region region-id="1692" left="114" top="1881" width="52.5" height="10.5" align-x="122.25" align-y="1890" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="typing" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">kph [Enter]</p>
</text>
</region>
<region region-id="2037" left="240" top="1880.25" width="36" height="12" align-x="247.5" align-y="1890" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
</text>
</region>
<region region-id="1686" left="18" top="1900.5" width="96.75" height="33.75" align-x="57" align-y="1920" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:real>15</ml:real>
<ml:id xml:space="preserve">mph</ml:id>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:real>6.7056000000000013</ml:real>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="meter"></unitReference>
<unitReference unit="second" power-numerator="-1"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
</math>
<rendering item-idref="15"></rendering>
</region>
<region region-id="2034" left="240" top="1900.5" width="96" height="33.75" align-x="279" align-y="1920" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:real>15</ml:real>
<ml:id xml:space="preserve">mph</ml:id>
</ml:apply>
<ml:unitOverride>
<ml:apply>
<ml:div></ml:div>
<ml:id xml:space="preserve">km</ml:id>
<ml:id xml:space="preserve">hr</ml:id>
</ml:apply>
</ml:unitOverride>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:real>24.140160000000005</ml:real>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="hour" power-numerator="-1"></unitReference>
<unitReference unit="kilometer"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
</math>
<rendering item-idref="16"></rendering>
</region>
<region region-id="2035" left="240" top="1940.25" width="12" height="12" align-x="244.5" align-y="1950" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="false">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">or </p>
</text>
</region>
<region region-id="2041" left="240" top="1962.75" width="96" height="15.75" align-x="279" align-y="1974" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:real>15</ml:real>
<ml:id xml:space="preserve">mph</ml:id>
</ml:apply>
<ml:unitOverride>
<ml:id xml:space="preserve">kph</ml:id>
</ml:unitOverride>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:real>24.140160000000005</ml:real>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="kilometers_per_hour"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
</math>
<rendering item-idref="17"></rendering>
</region>
<region region-id="1858" left="0" top="1986" width="6000" height="6" align-x="0" align-y="1986" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="1707" left="6" top="1994.25" width="193.5" height="12" align-x="6" align-y="2004" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To Define Your Own Units</p>
</text>
<indexes>
<phrase>units, defining custom</phrase>
</indexes>
</region>
<region region-id="1738" left="6" top="2018.25" width="159" height="24" align-x="17.25" align-y="2028" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To define a femtometer, which is not a built-in unit, type:</p>
</text>
</region>
<region region-id="2042" left="240" top="2030.25" width="36" height="12" align-x="247.5" align-y="2040" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
</text>
</region>
<region region-id="1844" left="6" top="2055" width="180.75" height="10.5" align-x="8.25" align-y="2064" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="typing" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">fm:10^-15 [Spacebar] [Spacebar] *m</p>
</text>
</region>
<region region-id="2029" left="240" top="2047.5" width="72.75" height="21" align-x="259.5" align-y="2064" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">fm</ml:id>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:apply>
<ml:pow></ml:pow>
<ml:real>10</ml:real>
<ml:real>-15</ml:real>
</ml:apply>
<ml:id xml:space="preserve">m</ml:id>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="18"></rendering>
</region>
<region region-id="1845" left="6" top="2076.75" width="27.75" height="14.25" align-x="6" align-y="2088" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Try it!</p>
</text>
</region>
<region region-id="1928" left="6" top="2112.75" width="227.25" height="42.75" align-x="12.75" align-y="2124" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<b>
<c val="#008080">Tip</c>
</b>: A unit can have any name you want to give it, but it is best to use accepted abbreviations when you can.</p>
</text>
</region>
<region region-id="1724" left="6" top="2174.25" width="207" height="36" align-x="12" align-y="2184" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To convert a length measure to femtometers using the unit you defined above, click the result below, then click the <inlineAttr font-weight="normal">units placeholder</inlineAttr> and type:</p>
</text>
</region>
<region region-id="1734" left="6" top="2235" width="49.5" height="10.5" align-x="12" align-y="2244" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="typing" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">fm [Enter]</p>
</text>
</region>
<region region-id="2044" left="240" top="2234.25" width="36" height="12" align-x="240" align-y="2244" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
</text>
</region>
<region region-id="1727" left="24" top="2257.5" width="142.5" height="21" align-x="86.25" align-y="2274" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:real>15</ml:real>
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:apply>
<ml:pow></ml:pow>
<ml:real>10</ml:real>
<ml:real>-10</ml:real>
</ml:apply>
<ml:id xml:space="preserve">m</ml:id>
</ml:apply>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:real>1.5E-09</ml:real>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="meter"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
</math>
<rendering item-idref="19"></rendering>
</region>
<region region-id="2048" left="240" top="2257.5" width="122.25" height="21" align-x="302.25" align-y="2274" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:real>15</ml:real>
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:apply>
<ml:pow></ml:pow>
<ml:real>10</ml:real>
<ml:real>-10</ml:real>
</ml:apply>
<ml:id xml:space="preserve">m</ml:id>
</ml:apply>
</ml:apply>
<ml:unitOverride>
<ml:id xml:space="preserve">fm</ml:id>
</ml:unitOverride>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:real>1499999.9999999998</ml:real>
</result>
</ml:eval>
</math>
<rendering item-idref="20"></rendering>
</region>
<region region-id="1736" left="0" top="2298" width="6000" height="6" align-x="0" align-y="2298" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="1750" left="6" top="2312.25" width="159" height="24" align-x="17.25" align-y="2322" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To define a unit that can be used to track points gained or lost, type:</p>
</text>
</region>
<region region-id="2049" left="240" top="2324.25" width="36" height="12" align-x="247.5" align-y="2334" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
</text>
</region>
<region region-id="1946" left="6" top="2349" width="19.5" height="10.5" align-x="8.25" align-y="2358" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="typing" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">pt:1</p>
</text>
</region>
<region region-id="2050" left="240" top="2346.75" width="33.75" height="15.75" align-x="257.25" align-y="2358" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">pt</ml:id>
<ml:real>1</ml:real>
</ml:define>
</math>
<rendering item-idref="21"></rendering>
</region>
<region region-id="1754" left="6" top="2382.75" width="27.75" height="14.25" align-x="6" align-y="2394" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Try it!</p>
</text>
</region>
<region region-id="2051" left="6" top="2418.75" width="294.75" height="42.75" align-x="6" align-y="2430" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<b>
<c val="#008080">Tip</c>
</b>:You can define any unit to have the value 1 in order to display that label in results. The unit doesn't need to have a pre-existing definition.</p>
</text>
</region>
<region region-id="1760" left="6" top="2486.25" width="217.5" height="36" align-x="12" align-y="2496" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To display that unit in a result, such as the one below, click the result, then click the<b>
<sp></sp>
</b>
<inlineAttr font-weight="normal">units placeholder</inlineAttr> and type:</p>
</text>
</region>
<region region-id="1774" left="6" top="2535" width="61.5" height="10.5" align-x="12" align-y="2544" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="typing" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">pt/mi [Enter]</p>
</text>
</region>
<region region-id="1770" left="18" top="2562.75" width="102.75" height="15.75" align-x="87.75" align-y="2574" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">Δp_wingspan</ml:id>
<ml:apply>
<ml:mult style="dot"></ml:mult>
<ml:real>2</ml:real>
<ml:id xml:space="preserve">pt</ml:id>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="22"></rendering>
</region>
<region region-id="1771" left="18" top="2586.75" width="106.5" height="15.75" align-x="90" align-y="2598" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">Δm_wingspan</ml:id>
<ml:apply>
<ml:mult style="dot"></ml:mult>
<ml:real>1</ml:real>
<ml:id xml:space="preserve">mi</ml:id>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="23"></rendering>
</region>
<region region-id="1772" left="18" top="2608.5" width="150.75" height="33.75" align-x="91.5" align-y="2628" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">rate_wingspan</ml:id>
<ml:apply>
<ml:div></ml:div>
<ml:id xml:space="preserve">Δp_wingspan</ml:id>
<ml:id xml:space="preserve">Δm_wingspan</ml:id>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="24"></rendering>
</region>
<region region-id="2052" left="240" top="2618.25" width="36" height="12" align-x="240" align-y="2628" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
</text>
</region>
<region region-id="2269" left="240" top="2646" width="108" height="38.25" align-x="240" align-y="2646" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<picture>
<png item-idref="25" display-width="106.5" display-height="36.75"></png>
</picture>
<rendering item-idref="26"></rendering>
</region>
<region region-id="1773" left="18" top="2650.5" width="130.5" height="33.75" align-x="90.75" align-y="2670" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:id xml:space="preserve">rate_wingspan</ml:id>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:real>0.0012427423844746676</ml:real>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="meter" power-numerator="-1"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
</math>
<rendering item-idref="27"></rendering>
</region>
<region region-id="1798" left="0" top="2706" width="6000" height="6" align-x="0" align-y="2706" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="1785" left="6" top="2720.25" width="255" height="12" align-x="6" align-y="2730" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To Change the Unit System of the Worksheet</p>
</text>
<indexes>
<phrase>unit system, changing</phrase>
<phrase>changing the unit system</phrase>
</indexes>
</region>
<region region-id="1786" left="6" top="2744.25" width="274.5" height="12" align-x="17.25" align-y="2754" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Select<b> Worksheet Options</b> from the <b>Tools</b> menu.</p>
</text>
</region>
<region region-id="2219" left="6" top="2768.25" width="257.25" height="36" align-x="17.25" align-y="2778" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Click the <b>Unit System</b> tab, and select U.S. This option gives you results in feet, inches, and other "English System" measures. Then click OK.</p>
</text>
</region>
<region region-id="2121" left="6" top="2822.25" width="92.25" height="12" align-x="12" align-y="2832" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To test this out, type:</p>
</text>
</region>
<region region-id="2122" left="6" top="2847" width="43.5" height="10.5" align-x="8.25" align-y="2856" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="typing" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">6ft+2m=</p>
</text>
</region>
<region region-id="2123" left="6" top="2870.25" width="100.5" height="12" align-x="15" align-y="2880" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You get a result in feet.</p>
</text>
</region>
<region region-id="2124" left="6" top="2898.75" width="37.5" height="29.25" align-x="6" align-y="2910" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Try it! <f family="Symbol" charset="2">ß</f>
</p>
</text>
</region>
<region region-id="2261" left="144" top="2930.25" width="36" height="12" align-x="144" align-y="2940" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
</text>
</region>
<region region-id="2272" left="144" top="2952" width="107.25" height="18.75" align-x="144" align-y="2952" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<picture>
<png item-idref="28" display-width="105.75" display-height="17.25"></png>
</picture>
<rendering item-idref="29"></rendering>
</region>
<region region-id="1980" left="6" top="2982.75" width="309" height="14.25" align-x="17.25" align-y="2994" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="false">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Now change the unit system back to SI in the same Unit System tab.</p>
</text>
</region>
<region region-id="2259" left="144" top="3008.25" width="36" height="12" align-x="144" align-y="3018" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
</text>
</region>
<region region-id="2258" left="144" top="3024" width="103.5" height="20.25" align-x="144" align-y="3024" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<picture>
<png item-idref="30" display-width="102" display-height="18.75"></png>
</picture>
<rendering item-idref="31"></rendering>
</region>
<region region-id="2263" left="6" top="3054.75" width="338.25" height="71.25" align-x="6" align-y="3066" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To set your unit system to return a customized unit for any quantity, open the Unit System tab and choose the Custom system. Base the system around a standard, such as SI, then modify the measure of interest. For example, you can choose the U.S. system, but return all lengths in inches. Make these modifications and look at the results in the above example. </p>
</text>
</region>
<region region-id="1898" left="6" top="3140.25" width="210.75" height="12" align-x="27" align-y="3150" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To Check Your Calculations with Units</p>
</text>
</region>
<region region-id="1899" left="6" top="3156.75" width="341.25" height="57" align-x="6" align-y="3168" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The units feature is also very convenient because it lets you know if you have made a mistake in the units in your calculations. For example, the result of the force calculation below should be in <b>newtons</b> (or an equivalent force unit):</p>
</text>
</region>
<region region-id="1880" left="36" top="3220.5" width="57.75" height="33.75" align-x="49.5" align-y="3240" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">a</ml:id>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:real>10</ml:real>
<ml:apply>
<ml:div></ml:div>
<ml:id xml:space="preserve">m</ml:id>
<ml:id xml:space="preserve">sec</ml:id>
</ml:apply>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="32"></rendering>
</region>
<region region-id="2257" left="144" top="3228.75" width="62.25" height="15.75" align-x="174.75" align-y="3240" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">mass</ml:id>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:real>2</ml:real>
<ml:id xml:space="preserve">kg</ml:id>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="33"></rendering>
</region>
<region region-id="1882" left="36" top="3270.75" width="57" height="15.75" align-x="51" align-y="3282" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">F</ml:id>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:id xml:space="preserve">mass</ml:id>
<ml:id xml:space="preserve">a</ml:id>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="34"></rendering>
</region>
<region region-id="2256" left="144" top="3262.5" width="72" height="33.75" align-x="158.25" align-y="3282" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:id xml:space="preserve">F</ml:id>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:real>20</ml:real>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="kilogram"></unitReference>
<unitReference unit="meter"></unitReference>
<unitReference unit="second" power-numerator="-1"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
</math>
<rendering item-idref="35"></rendering>
</region>
<region region-id="1988" left="0" top="3306" width="6000" height="6" align-x="0" align-y="3306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="1884" left="6" top="3306" width="341.25" height="52.5" align-x="6" align-y="3318" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">but it appears to have an extraneous factor of <region region-id="3" left="213" top="3306.75" width="37.5" height="15.75" align-x="231" align-y="3318" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:id xml:space="preserve">seconds</ml:id>
</math>
<rendering item-idref="36"></rendering>
</region>. Looking closely at the statements in the calculation, you can see that the definition of the acceleration <region region-id="827" left="63" top="3342.75" width="6" height="15.75" align-x="65.25" align-y="3354" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:id xml:space="preserve">a</ml:id>
</math>
<rendering item-idref="37"></rendering>
</region> is missing a term of <region region-id="828" left="161.25" top="3337.5" width="29.25" height="21" align-x="174.75" align-y="3354" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:apply>
<ml:pow></ml:pow>
<ml:id xml:space="preserve">sec</ml:id>
<ml:real>-1</ml:real>
</ml:apply>
</math>
<rendering item-idref="38"></rendering>
</region>.</p>
</text>
</region>
<region region-id="1886" left="6" top="3386.25" width="160.5" height="12" align-x="6" align-y="3396" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<b>
<c val="#800000">To Format Displayed Units</c>
</b>
</p>
</text>
<indexes>
<phrase>formatting displayed units</phrase>
<phrase>units, formatting</phrase>
</indexes>
</region>
<region region-id="1887" left="6" top="3402.75" width="341.25" height="57" align-x="6" align-y="3414" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To display units choose <b>Result</b> under the <b>Format</b> menu. Under the heading "Unit Display," check "Format units" and "Simplify units when possible." Units in your results display as shown below. These options can be set as the default or for individual equations.</p>
</text>
</region>
<region region-id="2088" left="6" top="3468.75" width="326.25" height="28.5" align-x="12" align-y="3480" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="true" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To set a Unit Display option as the default for a worksheet, click anywhere in the worksheet and choose <b>Result</b> from the <b>Format</b> menu.</p>
</text>
</region>
<region region-id="2090" left="6" top="3504.75" width="337.5" height="28.5" align-x="12" align-y="3516" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="true" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To set one of these Display options for a single result, click on the result, then choose <b>Result</b> from the <b>Format</b> menu.</p>
</text>
</region>
<region region-id="1888" left="12" top="3547.5" width="89.25" height="21" align-x="24.75" align-y="3564" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:id xml:space="preserve">g</ml:id>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:real>9.80665</ml:real>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="meter"></unitReference>
<unitReference unit="second" power-numerator="-2"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
<resultFormat>
<general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"></general>
<matrix display-style="auto" expand-nested-arrays="false"></matrix>
<unit format-units="false" simplify-units="false" fractional-unit-exponent="false"></unit>
</resultFormat>
</math>
<rendering item-idref="39"></rendering>
</region>
<region region-id="2274" left="150" top="3552.75" width="96.75" height="14.25" align-x="150" align-y="3564" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="teal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Units not formatted</p>
</text>
</region>
<region region-id="1890" left="12" top="3586.5" width="72" height="39" align-x="24.75" align-y="3606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:id xml:space="preserve">g</ml:id>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:real>9.80665</ml:real>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="meter"></unitReference>
<unitReference unit="second" power-numerator="-2"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
</math>
<rendering item-idref="40"></rendering>
</region>
<region region-id="2275" left="150" top="3594.75" width="78" height="14.25" align-x="150" align-y="3606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="teal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Units formatted</p>
</text>
</region>
<region region-id="2127" left="12" top="3629.25" width="134.25" height="44.25" align-x="70.5" align-y="3654" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:div></ml:div>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:real>5</ml:real>
<ml:id xml:space="preserve">kg</ml:id>
</ml:apply>
<ml:real>4</ml:real>
</ml:apply>
<ml:apply>
<ml:pow></ml:pow>
<ml:id xml:space="preserve">m</ml:id>
<ml:real>2</ml:real>
</ml:apply>
</ml:apply>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:real>20</ml:real>
<ml:apply>
<ml:pow></ml:pow>
<ml:id xml:space="preserve">s</ml:id>
<ml:real>2</ml:real>
</ml:apply>
</ml:apply>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:real>1</ml:real>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="kilogram"></unitReference>
<unitReference unit="meter" power-numerator="2"></unitReference>
<unitReference unit="second" power-numerator="-2"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
<resultFormat>
<general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"></general>
<matrix display-style="auto" expand-nested-arrays="false"></matrix>
<unit format-units="false" simplify-units="false" fractional-unit-exponent="false"></unit>
</resultFormat>
</math>
<rendering item-idref="41"></rendering>
</region>
<region region-id="2276" left="150" top="3648.75" width="90.75" height="14.25" align-x="150" align-y="3660" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="teal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Default unit result</p>
</text>
</region>
<region region-id="2128" left="12" top="3689.25" width="84.75" height="44.25" align-x="66.75" align-y="3714" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:div></ml:div>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:real>5</ml:real>
<ml:id xml:space="preserve">kg</ml:id>
</ml:apply>
<ml:real>4</ml:real>
</ml:apply>
<ml:apply>
<ml:pow></ml:pow>
<ml:id xml:space="preserve">m</ml:id>
<ml:real>2</ml:real>
</ml:apply>
</ml:apply>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:real>20</ml:real>
<ml:apply>
<ml:pow></ml:pow>
<ml:id xml:space="preserve">s</ml:id>
<ml:real>2</ml:real>
</ml:apply>
</ml:apply>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:real>1</ml:real>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="joule"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
</math>
<rendering item-idref="42"></rendering>
</region>
<region region-id="2277" left="150" top="3702.75" width="77.25" height="14.25" align-x="150" align-y="3714" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="teal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Units simplified</p>
</text>
</region>
<region region-id="2100" left="6" top="3752.25" width="180.75" height="12" align-x="6" align-y="3762" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Units in Range Variables</p>
</text>
<indexes>
<phrase>range variables</phrase>
<phrase>dimensions in range variables</phrase>
</indexes>
</region>
<region region-id="2101" left="6" top="3774.75" width="319.5" height="71.25" align-x="6" align-y="3786" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">
<b>
<u>
<link href=".\..\Features\feat_03.xmcd" title="Links to section" popup="false">Range variables</link>
</u>
</b> that include units must specifically include the stepsize for the range; automatically chosen step sizes would assume the base unit of the chosen worksheet system, resulting, for example, in stepping from 0 cm to 1 meter in the SI system of units. The required notation forces you to be clear.</p>
</text>
</region>
<region region-id="2131" left="6" top="3860.25" width="247.5" height="12" align-x="12" align-y="3870" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To define a range variable with dimensions, type:</p>
</text>
</region>
<region region-id="2151" left="12" top="3879" width="174" height="10.5" align-x="14.25" align-y="3888" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="typing" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">r:0cm,1cm;10cm</p>
</text>
</region>
<region region-id="2140" left="0" top="3900" width="6000" height="6" align-x="0" align-y="3900" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="2149" left="6" top="3908.25" width="247.5" height="12" align-x="12" align-y="3918" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To see the values in the range, type:</p>
</text>
</region>
<region region-id="2152" left="12" top="3927" width="174" height="10.5" align-x="14.25" align-y="3936" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="typing" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">r=</p>
</text>
</region>
<region region-id="2144" left="6" top="3954.75" width="37.5" height="29.25" align-x="6" align-y="3966" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Try it! <f family="Symbol" charset="2">ß</f>
</p>
</text>
</region>
<region region-id="2145" left="156" top="3956.25" width="36" height="12" align-x="156" align-y="3966" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
</text>
</region>
<region region-id="2146" left="156" top="3978.75" width="96.75" height="15.75" align-x="168" align-y="3990" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:id xml:space="preserve">r</ml:id>
<ml:range>
<ml:sequence>
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:real>0</ml:real>
<ml:id xml:space="preserve">cm</ml:id>
</ml:apply>
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:real>1</ml:real>
<ml:id xml:space="preserve">cm</ml:id>
</ml:apply>
</ml:sequence>
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:real>5</ml:real>
<ml:id xml:space="preserve">cm</ml:id>
</ml:apply>
</ml:range>
</ml:define>
</math>
<rendering item-idref="43"></rendering>
</region>
<region region-id="2147" left="156" top="4002.75" width="55.5" height="124.5" align-x="157.5" align-y="4014" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:id xml:space="preserve">r</ml:id>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:matrix rows="6" cols="1">
<ml:real>0</ml:real>
<ml:real>0.01</ml:real>
<ml:real>0.02</ml:real>
<ml:real>0.03</ml:real>
<ml:real>0.04</ml:real>
<ml:real>0.05</ml:real>
</ml:matrix>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="meter"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
<resultFormat>
<general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"></general>
<matrix display-style="auto" expand-nested-arrays="false"></matrix>
<unit format-units="false" simplify-units="false" fractional-unit-exponent="false"></unit>
<table item-idref="44"></table>
</resultFormat>
</math>
<rendering item-idref="45"></rendering>
</region>
<region region-id="2153" left="6" top="4116.75" width="295.5" height="28.5" align-x="6" align-y="4128" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Ranges with units are particularly useful for <b>
<u>
<link href=".\..\..\Visualization\unitsplt.xmcd" title="Links to section" popup="false">plotting</link>
</u>
</b> functions that take units.</p>
</text>
</region>
<region region-id="1967" left="6" top="4160.25" width="176.25" height="12" align-x="6" align-y="4170" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Heading 2" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Functions that Accept Units</p>
</text>
</region>
<region region-id="2154" left="6" top="4176.75" width="310.5" height="28.5" align-x="6" align-y="4188" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Since functions can do any of the above calculations, functions also balance units. </p>
</text>
</region>
<region region-id="2173" left="6" top="4214.25" width="24.75" height="12" align-x="17.25" align-y="4224" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Type:</p>
</text>
</region>
<region region-id="2185" left="12" top="4233" width="74.25" height="10.5" align-x="12" align-y="4242" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="typing" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">d(x,v):3x+v*sec</p>
</text>
</region>
<region region-id="2192" left="6" top="4256.25" width="247.5" height="12" align-x="12" align-y="4266" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">To evaluate the function, type:</p>
</text>
</region>
<region region-id="2193" left="12" top="4275" width="174" height="10.5" align-x="14.25" align-y="4284" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="typing" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">d(2*cm,3*cm/sec)=</p>
</text>
</region>
<region region-id="2216" left="6" top="4296.75" width="37.5" height="29.25" align-x="6" align-y="4308" show-border="false" show-highlight="true" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="0" text-align="inherit" list-style-type="none" tabs="inherit">Try it! <f family="Symbol" charset="2">ß</f>
</p>
</text>
</region>
<region region-id="2217" left="180" top="4298.25" width="20.25" height="12" align-x="187.5" align-y="4308" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="directions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">See:</p>
</text>
</region>
<region region-id="2188" left="180" top="4314.75" width="99.75" height="15.75" align-x="220.5" align-y="4326" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:function>
<ml:id xml:space="preserve">d</ml:id>
<ml:boundVars>
<ml:id xml:space="preserve">x</ml:id>
<ml:id xml:space="preserve">v</ml:id>
</ml:boundVars>
</ml:function>
<ml:apply>
<ml:plus></ml:plus>
<ml:apply>
<ml:mult style="auto-select"></ml:mult>
<ml:real>3</ml:real>
<ml:id xml:space="preserve">x</ml:id>
</ml:apply>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:id xml:space="preserve">v</ml:id>
<ml:id xml:space="preserve">sec</ml:id>
</ml:apply>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="46"></rendering>
</region>
<region region-id="2189" left="180" top="4342.5" width="133.5" height="33.75" align-x="264.75" align-y="4362" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:id xml:space="preserve">d</ml:id>
<ml:sequence>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:real>2</ml:real>
<ml:id xml:space="preserve">cm</ml:id>
</ml:apply>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:real>3</ml:real>
<ml:apply>
<ml:div></ml:div>
<ml:id xml:space="preserve">cm</ml:id>
<ml:id xml:space="preserve">sec</ml:id>
</ml:apply>
</ml:apply>
</ml:sequence>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<unitedValue>
<ml:real>0.09</ml:real>
<unitMonomial xmlns="http://schemas.mathsoft.com/units10">
<unitReference unit="meter"></unitReference>
</unitMonomial>
</unitedValue>
</result>
</ml:eval>
</math>
<rendering item-idref="47"></rendering>
</region>
<region region-id="2200" left="6" top="4392.75" width="311.25" height="28.5" align-x="6" align-y="4404" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Mathcad gives you similar errors if you try to call a function with units that don't make sense.</p>
</text>
</region>
<region region-id="2206" left="12" top="4434.75" width="96.75" height="15.75" align-x="87.75" align-y="4446" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math error="This value has units: Length, &#xA;but must have units: Length · Time^-1." optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:id xml:space="preserve">d</ml:id>
<ml:sequence>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:real>2</ml:real>
<ml:id xml:space="preserve">cm</ml:id>
</ml:apply>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:real>3</ml:real>
<ml:id xml:space="preserve">cm</ml:id>
</ml:apply>
</ml:sequence>
</ml:apply>
</ml:eval>
</math>
<metadata>
<customValue name="Mathsoft_expected_error" type="text">QASET</customValue>
</metadata>
<rendering item-idref="48"></rendering>
</region>
<region region-id="2202" left="0" top="4458" width="6000" height="6" align-x="0" align-y="4458" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<pageBreak></pageBreak>
</region>
<region region-id="2199" left="6" top="4464.75" width="302.25" height="42.75" align-x="6" align-y="4476" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">If you want to know what kind of arguments a function expects without altering values, you can look at the function signature by evaluating the name of the function.</p>
</text>
</region>
<region region-id="2207" left="12" top="4518.75" width="195" height="15.75" align-x="25.5" align-y="4530" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:id xml:space="preserve">d</ml:id>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:id xml:space="preserve">ƒ(any1 · Time, any1) → any1 · Time</ml:id>
</result>
</ml:eval>
</math>
<rendering item-idref="49"></rendering>
</region>
<region region-id="2214" left="6" top="4542.75" width="306" height="71.25" align-x="6" align-y="4554" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">What this says is "The name d is a function of two arguments. The first argument can have any dimensions, the second argument has the dimensions of the first divided by time. The result has the same dimensions as the first argument." The compact notation allows Mathcad to express the signature of any type of function.</p>
</text>
</region>
<region region-id="2266" left="18" top="4627.5" width="75.75" height="21" align-x="72.75" align-y="4644" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:define>
<ml:function>
<ml:id xml:space="preserve">pow</ml:id>
<ml:boundVars>
<ml:id xml:space="preserve">x</ml:id>
<ml:id xml:space="preserve">n</ml:id>
</ml:boundVars>
</ml:function>
<ml:apply>
<ml:pow></ml:pow>
<ml:id xml:space="preserve">x</ml:id>
<ml:id xml:space="preserve">n</ml:id>
</ml:apply>
</ml:define>
</math>
<rendering item-idref="50"></rendering>
</region>
<region region-id="2211" left="120" top="4632.75" width="216" height="57" align-x="123" align-y="4644" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit">
<b>
<c val="#008080">Tip:</c>
</b> It is not possible to define a function that returns units that depend on the value of the inputs; it cannot be unit balanced without knowing the particular value of the inputs.</p>
</text>
</region>
<region region-id="2265" left="18" top="4662.75" width="88.5" height="15.75" align-x="85.5" align-y="4674" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math error="This value has units: Length, &#xA;but must have units: Unitless." optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:id xml:space="preserve">pow</ml:id>
<ml:sequence>
<ml:apply>
<ml:mult style="default"></ml:mult>
<ml:real>2</ml:real>
<ml:id xml:space="preserve">m</ml:id>
</ml:apply>
<ml:real>3</ml:real>
</ml:sequence>
</ml:apply>
</ml:eval>
</math>
<metadata>
<customValue name="Mathsoft_expected_error" type="text">bad_dimensions	Unitless	Length</customValue>
</metadata>
<rendering item-idref="51"></rendering>
</region>
<region region-id="1974" left="6" top="4698" width="317.25" height="73.5" align-x="6" align-y="4710" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<text use-page-width="false" push-down="false" lock-width="true">
<p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The error arises because the value of <region region-id="2218" left="174.75" top="4698.75" width="6" height="15.75" align-x="177" align-y="4710" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<math optimize="false" disable-calc="false">
<ml:id xml:space="preserve">n</ml:id>
</math>
<rendering item-idref="52"></rendering>
</region> determines the units of the output. Since Mathcad checks unit balancing before calculation begins on your document, it balances all the units without processing the particular arguments of the functions, and this kind of construct is not allowed. You can call such a function without units:</p>
</text>
</region>
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<math optimize="false" disable-calc="false">
<ml:eval placeholderMultiplicationStyle="default" resultMultiplicationStyle="default">
<ml:apply>
<ml:id xml:space="preserve">pow</ml:id>
<ml:sequence>
<ml:real>2</ml:real>
<ml:real>3</ml:real>
</ml:sequence>
</ml:apply>
<result xmlns="http://schemas.mathsoft.com/math30">
<ml:real>8</ml:real>
</result>
</ml:eval>
</math>
<rendering item-idref="53"></rendering>
</region>
<region region-id="2273" left="6" top="4824" width="426.75" height="5.25" align-x="6" align-y="4824" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag="">
<picture>
<png item-idref="54" display-width="425.25" display-height="3.75"></png>
</picture>
<rendering item-idref="55"></rendering>
</region>
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