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 50 yards. Through what angle should the sprinkler rotate?
Hello, magentarita!
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 50 yards.
Through what angle should the sprinkler rotate?
We are expected to know the area formula for a sector of a circle: .$\displaystyle A \:=\:\tfrac{1}{2}r^2\theta$ .[1]
. . where $\displaystyle r$ is the radius and $\displaystyle \theta$ is the central angle in radians.
We know: .$\displaystyle A = 100,\;r = 50$
Substitute into [1]: .$\displaystyle 100 \:=\:\tfrac{1}{2}(50^2)\theta$
Therefore: .$\displaystyle \theta \;=\;0.08\text{ radians} \;\approx\;4.58^o$