Question

the diameter of a circle is 10 feet. what is the angle measure of an arc bounding a sector with area \\( \pi \\) square feet?
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. Given \( d = 10\) ft, so \( r=\frac{d}{2}=\frac{10}{2} = 5\) ft.

Step2: Use 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 know that \( K = \pi\) and \( r = 5\). Substitute these values into the formula:
\(\pi=\frac{\theta}{360}\times\pi\times(5)^{2}\)

Step3: Solve for \(\theta\)

First, divide both sides of the equation \(\pi=\frac{\theta}{360}\times\pi\times25\) by \(\pi\) (since \(\pi
eq0\)). We get \(1=\frac{\theta}{360}\times25\).
Then, solve for \(\theta\): \(\theta=\frac{360}{25}=\frac{72}{5} = 14.4\)

Answer:

\(14.4\)