Chapter 7.1, Problem 33E

(a)

To determine

The radius of the city.

Answer to Problem 33E

The radius of the city is $2\text{ miles}$ .

Explanation of Solution

Given:

The population density of the city approaches zero at the end of the city.

Calculation:

  $\begin{array}{l}P(r)=10,000(2-r)\\ P(r)=0\\ \end{array}$

Hence, the radius of the city is:

  $\begin{array}{l}0=2-r\\ r=2\text{ miles}\end{array}$

Conclusion:

The radius of the city is $2\text{ miles}$ .

(b)

To determine

The area of a ring.

Answer to Problem 33E

The area of a ring is $2\prod r\cdot \Delta r$

Explanation of Solution

Given:

The thickness of the ring is $\Delta r$ and radius is r.

Calculation:

When the ring is straighten then, it forms rectangular strip. Then, the area is calculated as:

  $\text{Area}=\text{ length}\times \text{breadth}$

  $\text{Area}=2\prod r\cdot \Delta r$

Conclusion:

The area of a ring is $2\prod r\cdot \Delta r$

(c)

To determine

The reason of population density being approximately $10,000(2-r)(2\prod r)\Delta r$ .

Answer to Problem 33E

The formula of population density is density multiplied by area. Therefore, population density is approximately $10,000(2-r)(2\prod r)\Delta r$

Explanation of Solution

Given:

The population density function is $10,000(2-r)$ .

Calculation:

The formula of population density is as follows:

  $\begin{array}{l}\partial P=Density\times Area\\ \partial P=10,000(2-r)(2\prod r)\Delta r\end{array}$

Conclusion:

The formula of population density is density multiplied by area. Therefore, population density is approximately $10,000(2-r)(2\prod r)\Delta r$

(d)

To determine

The total population of Washerton.

Answer to Problem 33E

The total population of Washerton is $83,733$ .

Explanation of Solution

Given:

The population density is approximately $10,000(2-r)(2\prod r)\Delta r$

Calculation:

  $\begin{array}{l}{\displaystyle \underset{0}{\overset{P}{\int }}d}P=(10,000)(2\prod ){\displaystyle \underset{0}{\overset{2}{\int }}(2-r)(r)dr}\\ P=10,000(2\prod ){\left[\frac{2r^2}{2}-\frac{r^3}{3}\right]}_0^2\\ P=10,000(2\prod )\left[\frac{8}{2}-\frac{8}{3}\right]\\ P=83,733\end{array}$

Conclusion:

The total population of Washerton is $83,733$ .