Question

find the area of the sector of a circle of radius 8 ft with a central angle of 150°. round the solution to two decimal places.

Explanation:

Step1: Convert angle to radians

The formula to convert degrees to radians is $\theta_{rad}=\theta_{deg}\times\frac{\pi}{180}$. So, $\theta = 150\times\frac{\pi}{180}=\frac{5\pi}{6}$ radians.

Step2: Use the sector - area formula

The formula for the area of a sector of a circle is $A=\frac{1}{2}r^{2}\theta$, where $r$ is the radius and $\theta$ is the central angle in radians. Given $r = 8$ ft and $\theta=\frac{5\pi}{6}$, we substitute these values into the formula: $A=\frac{1}{2}\times8^{2}\times\frac{5\pi}{6}$.

Step3: Simplify the expression

First, calculate $8^{2}=64$. Then, $A=\frac{1}{2}\times64\times\frac{5\pi}{6}=32\times\frac{5\pi}{6}=\frac{160\pi}{6}=\frac{80\pi}{3}$.

Step4: Round the result

Using $\pi\approx3.14159$, we have $A=\frac{80\times3.14159}{3}\approx83.78$.

Answer:

$83.78$ ft²