Q. 96 - Precalculus Enhanced with Graphing Utilities

Question

An engineer is asked to design a water sprinkler that will cover a field of $100$square yards that is in the shape of a sector of a circle of radius $15$ yards. Through what angle should the sprinkler rotate? 

Step-by-Step Solution

Answer

The sprinkler should rotate about $0.89$ radians or about $50.{93}^{\circ }$

Step 1. Given information

The field to be covered is of $100$ square yards and it is in the shape of a sector of the circle of radius $15$ yards.

Step 2. Used formulae

The area of a circle with radius $r$ and angle $\theta$ is given by, $$\begin{gathered} A= \frac{1}{2}{r}^2\theta \\ \theta =\frac{2A}{r^2} \end{gathered}$$

Step 3. Calculation

Substituting the values in the mentioned formula, we get, $\theta =\frac{2A}{r^2}=\frac{2·100}{{\left(15\right)}^2}=\frac{8}{9}\approx 0.89$ radians or $0.89·\frac{180}{\mathrm{\pi }}=50.{93}^{°}$