Question

the radius of a circle is 10 centimeters. what is the angle measure of an arc bounding a sector with area 17π square centimeters?
k=17π sq. cm
r=10 cm
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.
We are given that \(K = 17\pi\) and \(r = 10\). Substitute these values into the formula: \(17\pi=\frac{\theta}{360}\times\pi\times(10)^{2}\).

Step2: Simplify the equation

First, cancel out the \(\pi\) terms on both sides of the equation.
The equation \(17\pi=\frac{\theta}{360}\times\pi\times100\) becomes \(17=\frac{\theta\times100}{360}\).
Then, cross - multiply: \(17\times360 = 100\theta\).
So, \(6120=100\theta\).

Step3: Solve for \(\theta\)

Divide both sides of the equation \(6120 = 100\theta\) by 100.
\(\theta=\frac{6120}{100}=\frac{306}{5}=61.2\)

Answer:

\(61.2\)