Question

this exercise involves the formula for the area of a circular sector. a sector of a circle has a central angle of 115°. find the area of the sector if the radius of the circle is 7 ft. (round your answer to one decimal place.)

Explanation:

Step1: Convert angle to radians

First, convert the central - angle from degrees to radians. The conversion formula is $\theta_{rad}=\theta_{deg}\times\frac{\pi}{180}$. Given $\theta_{deg} = 115^{\circ}$, then $\theta_{rad}=115\times\frac{\pi}{180}=\frac{23\pi}{36}$ radians.

Step2: Use the area formula for a circular sector

The area formula for a circular sector is $A=\frac{1}{2}r^{2}\theta$, where $r$ is the radius of the circle and $\theta$ is the central - angle in radians. Here, $r = 7$ ft and $\theta=\frac{23\pi}{36}$ radians. So $A=\frac{1}{2}\times7^{2}\times\frac{23\pi}{36}$.

Step3: Calculate the area

$A=\frac{1}{2}\times49\times\frac{23\pi}{36}=\frac{1127\pi}{72}\approx48.8$ ft².

Answer:

$48.8$