Question

the radius of a circle is 12 inches. what is the angle measure of an arc bounding a sector with area 13π square inches?
give the exact answer in simplest form.

Explanation:

Step1: Recall the formula for the area of a sector

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 measure in degrees, and \(r\) is the radius of the circle.

Step2: Substitute the given values into the formula

We are given that \(K = 13\pi\) square inches and \(r = 12\) inches. Substituting these values into the formula \(K=\frac{\theta}{360}\times\pi r^{2}\), we get:
\(13\pi=\frac{\theta}{360}\times\pi\times(12)^{2}\)

Step3: Simplify the equation

First, cancel out the \(\pi\) terms on both sides of the equation. We have:
\(13=\frac{\theta}{360}\times144\)

Step4: Solve for \(\theta\)

Multiply both sides of the equation by \(360\) to get \(13\times360=\theta\times144\). Then \(\theta=\frac{13\times360}{144}\)
Simplify \(\frac{13\times360}{144}\):
\(\frac{13\times360}{144}=\frac{13\times360\div 72}{144\div 72}=\frac{13\times5}{2}=\frac{65}{2} = 32.5\)

Answer:

\(32.5\)