Question

in the figure, ( o ) is the center of the circle. the minor arc ( overarc{ab} ) has a length of ( 4pi ) and is ( \frac{1}{8} ) the circumference of the circle. if the area of the shaded region is ( api ), what is the value of ( a )?

Explanation:

Step1: Find the circumference of the circle

Since the length of arc \(\overset{\frown}{AB}\) is \(\frac{1}{8}\) of the circumference \(C\) of the circle, and \(l_{\overset{\frown}{AB}} = 4\pi\). Let \(C\) be the circumference. Then \(4\pi=\frac{1}{8}C\). Solving for \(C\), we get \(C = 32\pi\).

Step2: Find the radius of the circle

Using the formula \(C = 2\pi r\) (where \(r\) is the radius), substitute \(C = 32\pi\) into it. So \(32\pi=2\pi r\), and solving for \(r\) gives \(r = 16\).

Step3: Find the area of the circle

Using the formula \(A=\pi r^{2}\), substitute \(r = 16\). Then \(A=\pi\times16^{2}=256\pi\).

Step4: Find the area of the shaded region

Since the area of the shaded region is \(\frac{1}{8}\) of the area of the circle. Let \(A_{s}\) be the area of the shaded region. Then \(A_{s}=\frac{1}{8}\times256\pi = 32\pi\).

Answer:

\(32\)