Chapter 9.1, Problem 42E

(a)

To determine

To find: The radius of the region covered by the circular sprinkler if it covers an area of 200 square feet.

Answer to Problem 42E

The radius of the region covered by the sprinkler is $25.231$ feet.

Explanation of Solution

Given information:

A landscaper has installed a circular sprinkler that covers an area of $2000$ square feet.

Formula Used:

Use the formula for area of circle.

  $A=\pi r^2$

Calculation:

As given a circular sprinkler covers an area of $2000$ square feet. Substitute the area of the circle $2000$ square feet in the formula.

  $\begin{array}{c}\pi r^2=2000\\ r^2=\frac{2000}{\pi }\\ r^2=636.6197\\ r=\sqrt{636.6197}\\ r=25.231\end{array}$

Therefore, the radius of the region covered by the sprinkler is $25.231$ feet.

(b)

To determine

To find: The change in the radius when the area covered by the sprinkler is increased to $2500$ square feet.

Answer to Problem 42E

The change in the radius when the area covered by the sprinkler is increased to $2500$ square feet is $2.978$ feet.

Explanation of Solution

Given information:

A landscaper has installed a circular sprinkler that covers an area of $2500$ square feet.

Formula Used:

Use the formula for area of circle.

  $A=\pi r^2$

Calculation:

As calculated in part(a), the radius of the region covered by the sprinkler when the area is $2500$ square feet is $25.231$ feet. A circular sprinkler covers an area of $2500$ square feet.

Substitute the area of the circle $2500$ square feet in the formula.

  $\begin{array}{c}\pi r^2=2500\\ r^2=\frac{2500}{\pi }\\ r^2=795.7747\\ r=\sqrt{795.7747}\\ r=28.209\end{array}$

So, the radius of the region covered by the sprinkler when the area is $2500$ square feet is $28.209$ feet.

Calculate the change in the radius.

  $\begin{array}{c}\text{Change}=28.209-25.231\\ =2.978\text{feet}\end{array}$

Therefore, the change in the radius when the area covered by the sprinkler is increased to $2500$ square feet is $2.978$ feet.