Question

the diameter of a circle is 8 inches. what is the angle measure of an arc bounding a sector with area 6π square inches?
k=6π sq. in

d=8 in

give the exact answer in simplest form.

Explanation:

Step1: Find the radius of the circle

The formula for the radius \(r\) of a circle given the diameter \(d\) is \(r=\frac{d}{2}\). Since \(d = 8\) inches, then \(r=\frac{8}{2}=4\) inches.

Step2: Use the sector - area formula

The formula for the area of a sector of a circle is \(K=\frac{\theta}{360^{\circ}}\times\pi r^{2}\), 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 that \(K = 6\pi\) and \(r = 4\). Substitute these values into the formula:
\(6\pi=\frac{\theta}{360^{\circ}}\times\pi\times(4)^{2}\)
First, simplify the equation. Divide both sides of the equation by \(\pi\):
\(6=\frac{\theta}{360^{\circ}}\times16\)
Then, solve for \(\theta\). Multiply both sides by \(360^{\circ}\):
\(6\times360^{\circ}=16\theta\)
\(\theta=\frac{6\times360^{\circ}}{16}\)
Simplify the fraction \(\frac{6\times360^{\circ}}{16}=\frac{6\times45^{\circ}}{2}\) (dividing numerator and denominator by 8)
\(\theta = 135^{\circ}\)

Answer:

\(135\)