MathAdmin at 22:02, 26 May 2017

2017-05-26T22:02:17Z

MathAdmin: Created page with " A city bordered on one side by a lake can be approximated by a semicircle of radius 7 miles, whose city center is on the shoreline. As we move away from th..."

2017-04-10T16:49:21Z

Created page with " A city bordered on one side by a lake can be approximated by a semicircle of radius 7 miles, whose city center is on the shoreline. As we move away from th..."

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A city bordered on one side by a lake can be approximated by a semicircle of radius 7 miles, whose city center is on the shoreline. As we move away from the center along a radius the population density of the city can be approximated by:

::<math>\rho(x)=25000e^{-0.15x}</math>

people per square mile. What is the population of the city?

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'''Solution:'''

'''(a)'''

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'''(b)'''

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'''(c)'''

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[[009B_Sample_Final_2|'''Return to Sample Exam''']]

@@ Line 8: / Line 8: @@
 !Foundations: &nbsp;
 |-
+|Many word problems can be confusing, and this is a good example.
 |-
+|What changes, since we are only doing half of a disk?
 |-
+|Also, this particular problem will require integration by parts:
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp;<math>\int u\,dv=uv-\int v\,du.</math>
 |}
 '''Solution:'''
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Step 1: &nbsp;
 |-
+|We can treat this as a solid of revolution, and use the shell method.
 |-
+|We are working on a half disk of radius 7, so we can integrate a cross-section where &nbsp;<math style="vertical-align: 0px">x</math>&nbsp; goes from 0 to 7
 |-
+|and the height at each &nbsp;<math style="vertical-align: 0px">x</math>&nbsp; is our density function, &nbsp;<math style="vertical-align: -5px">\rho(x).</math>&nbsp;
 |-
+|Normally &nbsp;<math style="vertical-align: 0px">2\pi R</math>&nbsp; represents once around a circle of radius &nbsp;<math style="vertical-align: -5px">R,</math>&nbsp;
 |-
+|but in this case we only go half way around.
 |-
+|Therefore, we '''adjust''' our usual shell method formula to find the population as
 |-
+|
 |}
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-!Step 1: &nbsp;
+!Step 2: &nbsp;
 |-
+|Let's plug in the actual formula for density and solve. We have
 |-
+|
-|}
+&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
+\displaystyle{P} & = & \displaystyle{\int_{0}^{7}\pi x\cdot25,000e^{-0.15x}\,dx}\\
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+&&\\
-!Step 2: &nbsp;
+& = & \displaystyle{25,000\pi\int_{0}^{7}xe^{-0.15x}\,dx.}
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+|To solve this, we need to use integration by parts.
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+|Let &nbsp;<math style="vertical-align: 0px">u=x</math>&nbsp; and &nbsp;<math style="vertical-align: 0px">dv=e^{-0.15x}dx.</math>&nbsp;
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+|Then, &nbsp;<math style="vertical-align: 0px">du=dx</math>&nbsp; and &nbsp;<math style="vertical-align: -14px">v=-{\displaystyle \frac{e^{-0.15x}}{0.15}.}</math>&nbsp;
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+|Thus,
-|}
+&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
+\displaystyle{P} & = & \displaystyle{25,000\pi\int_{0}^{7}xe^{-0.15x}\,dx}\\
-{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
+&&\\
-!Step 2: &nbsp;
+& = & \displaystyle{25,000\pi\left[\left.-\frac{xe^{-0.15x}}{0.15}\right|_{0}^{7}+\int_{0}^{7}\frac{e^{-0.15x}}{0.15}\,dx\right]}\\
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+|Note that in a calculator-prohibited test, no one would expect the actual numerical answer.
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+|However, you would likely need the line above it to receive full credit.
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@@ Line 86: / Line 99: @@
 !Final Answer: &nbsp;
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+|&nbsp; &nbsp; &nbsp; &nbsp;<math>{\displaystyle 25,000\pi\left[\left(-\frac{7e^{-1.05}}{0.15}-\frac{e^{-1.05}}{(0.15)^{2}}\right)+\frac{1}{(0.15)^{2}}\right]\ \approx\ 986,556}</math>
 |}
 [[009B_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

009B Sample Final 2, Problem 4 - Revision history

MathAdmin at 22:02, 26 May 2017

MathAdmin: Created page with " A city bordered on one side by a lake can be approximated by a semicircle of radius 7 miles, whose city center is on the shoreline. As we move away from th..."