Chapter 9.1, Problem 92E

(a)

To determine

To find the radius of the region covered by the sprinkler

Answer to Problem 92E

The radius of the region covered by the sprinkler $r=25.231feet$

Explanation of Solution

Given:

Circular area covered by sprinkler =2000 square feet

Concept Used:

Area of circular region $=\pi r^2$ (where r is radius of circular region)

Using this formula find the radius of circular region

Calculation:

According to question

Area of circular region = 2000 square feet

Let r be radius of the region covered by the sprinkler

Using formula to find area circular region

  $\begin{array}{l}\pi r^2=2000\\ r^2=\frac{2000}{\pi }\\ \text{taking square root both sides}\\ \text{r=}\sqrt{\frac{2000}{\pi }}\\ r=25.231feet\end{array}$

Conclusion:

The radius of the region covered by the sprinkler $r=25.231feet$

(b)

To determine

To find how longer will be the radius if landscaper increases the area covered to 2500 square feet

Answer to Problem 92E

Increased length of Radius = $28.209-25.231=2.978feet$

Explanation of Solution

Given:

Increased area = 2500 square feet

Concept Used:

Find the radius corresponding to increased area.

Then,

Increased length of Radius = (Increased radius (R) $-$ Initial radius (r))

Calculation:

Let increased radius be R

According to question,

Increased area = 2500 square feet

Using formula to find area circular region

  $\begin{array}{l}\pi R^2=2500\\ R^2=\frac{2500}{\pi }\\ \text{taking square root both sides}\\ \text{R=}\sqrt{\frac{2500}{\pi }}\\ R=28.209feet\end{array}$

Conclusion:

Initial radius (r) = 25.231feet

Increased radius (R) = 28.209 feet

Increased length of Radius = $28.209-25.231=2.978feet$