Answer is: option4

\[ \int_0^8 \left(\frac{1}{4}(-x + 3)\right)\left(\frac{6}{\sqrt{x + 16}}\right) \, dx \]

Solution:

To find the total population of the city, we need to integrate the population density \( D(x) \) over the area of the city. The population density is given by:

\[ D(x) = \frac{6}{\sqrt{x + 16}} \]

where \( x \) is the distance from the river in miles, and \( D(x) \) is measured in thousands of people per square mile.

From the figure, the city is bounded by:

1. A river (one side).
2. A highway (opposite side).
3. The width of the city varies with distance \( x \) from the river.

The dimensions provided are:

1. The city extends from \( x = 0 \) (the river) to \( x = 8 \) miles (the highway).
2. At \( x = 0 \), the width is 3 miles.
3. At \( x = 8 \), the width is 5 miles.

Assuming the width \( w(x) \) changes linearly with \( x \), we can model it as:

\[ w(x) = \frac{1}{4}x + 3 \]

This is because:

1. At \( x = 0 \), \( w(0) = 3 \) miles.
2. At \( x = 8 \), \( w(8) = \frac{1}{4}(8) + 3 = 5 \) miles.

The total population \( P \) is the integral of the population density \( D(x) \) multiplied by the width \( w(x) \) over the length of the city from \( x = 0 \) to \( x = 8 \):

\[ P = \int_{0}^{8} w(x) \cdot D(x)\, dx = \int_{0}^{8} \left( \frac{1}{4}x + 3 \right) \cdot \frac{6}{\sqrt{x + 16}}\, dx \]

The correct integral matches option (D):