Question

the diameter of a circle is 40 feet. what is the angle measure of an arc bounding a sector with area ( 15pi ) square feet?
give the exact answer in simplest form.

Explanation:

Step1: Find the radius of the circle

The radius \(r\) of a circle is half of the diameter \(d\). Given \(d = 40\) feet, so \(r=\frac{d}{2}=\frac{40}{2}=20\) feet.

Step2: Use the sector - area formula

The formula for the area of a sector of a circle is \(K=\frac{\theta}{360}\times\pi r^{2}\), where \(K\) is the area of the sector, \(\theta\) is the central angle in degrees, and \(r\) is the radius of the circle.
We know \(K = 15\pi\) and \(r = 20\). Substitute these values into the formula:
\(15\pi=\frac{\theta}{360}\times\pi\times(20)^{2}\)
First, cancel out \(\pi\) on both sides of the equation: \(15=\frac{\theta}{360}\times400\)
Then, solve for \(\theta\):
\(\theta=\frac{15\times360}{400}\)
\(\theta=\frac{5400}{400}=\frac{27}{2}=13.5\)

Answer:

\(13.5\)