Population Density

brandon.boesch

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Jul 10, 2011
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If the population density of a particular city is given by p(r)= 20/(r+1) in thousands of people per square mile at a distance "r" miles from the center of the city, find the total number of people living within a 10 mile radius of the city center. (Hint: Notice that this is the density per unit area)

I got this problem wrong on my test and i'm not sure why. Here's what I did.
For this problem I tried taking the integral of the function from 0 to 10. This then gave me the population in thousands of people for 10 square mile (result : 20*ln(11) thousand people). With this I drew a box with each side having a length (20*ln(11))^(1/2) Inside this box I inscribed a circle with radius = ((20*ln(11))^(1/2)) / 2

To solve for the population I then used the equation P=(pi)r^2 with r=((20*ln(11))^(1/2)) / 2

Any ideas where I went wrong?
 

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The total area of consideraton is 100pi mi^2. If you drew something with a different radius, you have gone astray.

ROUGH estimates

High: 20*100*pi
Low: (20/11)*100*pi
Middle: (20/6)*100*pi

How does 20*ln(11) stack up?

While you are drawing, why not just draw a circle of radius 10 mi?
 
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