Answer is: option4
\( 2\pi \int_0^3 r (10 - 3\sqrt{r}) \, dr \)Solution:
Population in ring = \( 2\pi r \cdot f(r) \cdot dr \)
where:
- \( 2\pi r \) is the circumference of the ring,
- \( f(r) \) is the population density per square mile,
- \( dr \) is the thickness of the ring, making \( 2\pi r \, dr \) the area of that thin ring.
Given:
\[ f(r) = 10 - 3\sqrt{r} \]
Total population within radius 3 is:
\[ \text{Population} = \int_0^3 2\pi r \cdot (10 - 3\sqrt{r}) \, dr \]
This is:
\[ 2\pi \int_0^3 r(10 - 3\sqrt{r}) \, dr \]
Which exactly matches option (D).