Question

the area of the shaded sector is $8\pi$. what is the measure of $\angle sot$, which corresponds to the minor arc $st$?

Explanation:

Step1: Recall the formula for the area of a sector

The formula for the area of a sector of a circle is \(A=\frac{\theta}{360}\times\pi r^{2}\), where \(A\) is the area of the sector, \(\theta\) is the central angle (in degrees), and \(r\) is the radius of the circle.

Step2: Substitute the given values into the formula

We are given that \(A = 8\pi\) and \(r=8\). Substituting these values into the formula \(8\pi=\frac{\theta}{360}\times\pi\times8^{2}\).
First, simplify the equation. Divide both sides of the equation by \(\pi\): \(8=\frac{\theta}{360}\times64\).
Then, solve for \(\theta\). Multiply both sides by \(360\): \(8\times360 = 64\theta\).
So, \(2880=64\theta\).
Divide both sides by \(64\): \(\theta=\frac{2880}{64}=45\).

Answer:

B. \(45^{\circ}\)