Question

the radius of a circle is 4 meters. what is the angle measure of an arc bounding a sector with area 6π square meters?
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 \(K = 6\pi\) square meters and \(r = 4\) meters. Substituting these values into the formula \(K=\frac{\theta}{360}\times\pi r^{2}\), we get \(6\pi=\frac{\theta}{360}\times\pi\times(4)^{2}\).
First, simplify the right - hand side: \(\frac{\theta}{360}\times\pi\times16=\frac{16\pi\theta}{360}\).
Since \(\pi
eq0\), we can divide both sides of the equation \(6\pi=\frac{16\pi\theta}{360}\) by \(\pi\). The equation becomes \(6=\frac{16\theta}{360}\).

Step3: Solve for \(\theta\)

Cross - multiply: \(16\theta=6\times360\).
\(16\theta = 2160\).
Then \(\theta=\frac{2160}{16}\).
Simplify the fraction \(\frac{2160}{16}\) by dividing both the numerator and the denominator by 8. \(\frac{2160\div8}{16\div8}=\frac{270}{2}=135\).

Answer:

\(135\)