Answer is: option2

Solution:

We are given:

1. The population density function: \( D(x) = 15x\sqrt{x} - 3x^2 \)
2. This represents the number of people per square mile at a distance \( x \) miles from the highway.
3. The region is 11 miles wide, perpendicular to the highway.

We are to find the total number of people living between 16 and 25 miles from the highway.

To find the total population in the region from \( x = 16 \) to \( x = 25 \), we integrate the density function over that interval and multiply by the width (11 miles):

\[ \text{Population} = 11 \int_{16}^{25} \left(15x\sqrt{x} - 3x^2\right) dx \]

Note that \( 15x\sqrt{x} = 15x \cdot x^{1/2} = 15x^{3/2} \), so:

\[ \int_{16}^{25} \left(15x^{3/2} - 3x^2\right) dx \]

\[ \int \left(15x^{3/2} - 3x^2\right) dx = 15 \cdot \frac{2}{5}x^{5/2} - 3 \cdot \frac{1}{3}x^3 = 6x^{5/2} - x^3 \]

\[ \int_{16}^{25} \left(15x^{3/2} - 3x^2\right) dx = \left[6x^{5/2} - x^3\right]_{16}^{25} = 1077 \]

Multiply by 11 (the width of the region):

\[ \text{Total population} = 11 \cdot 1077 = \boxed{11,\!847} \]

Correct answer: (B) 11,847