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?

Short answer

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

Step-by-step solution

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$0.89·\frac{180}{\mathrm{\pi }}=50.{93}^{°}$