Question

the diameter of a circle is 16 inches. what is the angle measure of an arc bounding a sector with area 6π square inches?
k=6π sq. in

d=16 in

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 \(d = 16\) inches, so \(r=\frac{d}{2}=\frac{16}{2}=8\) inches.

Step2: Use the sector - area formula

The area formula of a sector 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.
We know \(K = 6\pi\) and \(r = 8\). Substitute these values into the formula:

$$6\pi=\frac{\theta}{360}\times\pi\times8^{2}$$

First, cancel out \(\pi\) on both sides of the equation:

$$6=\frac{\theta}{360}\times64$$

Then, solve for \(\theta\):
\(\theta=\frac{6\times360}{64}\)
Simplify the fraction \(\frac{6\times360}{64}=\frac{6\times45}{8}=\frac{270}{8}=\frac{135}{4}\)

Answer:

\(\frac{135}{4}\)