Question

1. in the diagram below of circle o, go=8, and m∠goj=60°. what is the area, in terms of π, of the shaded region? (1) \\(\frac{4π}{3}\\) (2) \\(\frac{20π}{3}\\) (3) \\(\frac{32π}{3}\\) (4) \\(\frac{160π}{3}\\)

Explanation:

Step1: Find total circle area

The formula for the area of a circle is $A = \pi r^2$. Here, radius $r=8$.
$A_{total} = \pi (8)^2 = 64\pi$

Step2: Find unshaded sector angle

The unshaded sector has a central angle of $60^\circ$. The shaded region's central angle is the total circle angle minus this:
$\theta_{shaded} = 360^\circ - 60^\circ = 300^\circ$

Step3: Calculate shaded area

The shaded area is the fraction $\frac{300}{360}$ of the total circle area.
$A_{shaded} = \frac{300}{360} \times 64\pi = \frac{5}{6} \times 64\pi = \frac{320\pi}{6} = \frac{160\pi}{3}$

Answer:

(4) $\frac{160\pi}{3}$