Population Density

[brandon.boesch]

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?

 

[Deleted member 4993]

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?

\(\displaystyle P \ = \ 4 * \int_0^{\frac{\pi}{2}}\int_0^{10}\frac{20}{r+1}(r\cdot dr)(d\theta) \ = \ 40 * \pi * \int_0^{10}\left [1 \ - \ \frac{1}{r+1}\right ]dr\)