Chapter 4.1, Problem 69E

To determine

To find: the area of the region and also draw the region.

Answer to Problem 69E

  

  $A\approx 294.52m^2$

Explanation of Solution

Given:

A sprinkler on a golf green is set to spray water over a distance of 15 meters and to rotate through an angle of $150°$ .

Concept Used:

Area of sector of circle:

For a circle of radius r, the area A of a sector of the circle with central angle $\theta$ (in radian measure)is:

  $A=\frac{1}{2}{r}^2\theta$

The diagram that shows the region that can be irrigated with the sprinkler is given below:

  

So, the area of the region that can be irrigated with the sprinkler is actually the area of a sector of circle with radius $r=15\text{ m}$ and central angle $\theta =150°$ .

First convert the central angle from degree measure to radian measure.

  $\begin{array}{c}\theta =150°\times \frac{\pi }{180°}\\ =150°\times \frac{\pi }{180°}\\ =150°\times \frac{\pi }{180°}\\ =\frac{5\pi }{6}\end{array}$

Thus, the area of the region is,

  $\begin{array}{c}A=\frac{1}{2}{r}^2\theta \\ =\frac{1}{2}\times {\left(15\right)}^2\times \left(\frac{5\pi }{6}\right)\\ \approx 294.52{\text{ m}}^2\end{array}$