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An engineer is asked to design a water sprinkler that will cover a field of 75 square yards that is in the shape of a sector of a circle of radius 12 yards. Through what angle should the sprinkler rotate?

Can somebody help me with the formula for this?

Trigonometry

Chelsrenee

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Answer 1

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The area of a circular sector with radius $r$ and angle $\theta$ (in radians) is
\[A=\frac{r^2 \theta}{2}.\]
In the given exaample we have $r=12$ and $A=75$. Substituting in the above formula we get
\[75=\frac{12^2 \theta}{2}=\frac{144 \theta}{2}=72\theta.\]
Hence
\[\theta =\frac{75}{72} \text{radians}.\]
If you want the angle in degrees then it is
\[\frac{75}{72} \times \frac{180}{\pi}=60.523 \text{degrees}.\]

Nirenberg

Nirenberg

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