Question

what is the area of a sector with a central angle of \frac{5\pi}{7} radians and a diameter of 5.6 in.? use 3.14 for \pi and round your final answer to the nearest hundredth. enter your answer as a decimal in the box.

Explanation:

Step1: Find the radius

The radius \(r\) is half of the diameter. Given \(d = 5.6\) in, so \(r=\frac{d}{2}=\frac{5.6}{2}=2.8\) in.

Step2: Use the sector - area formula

The formula for the area of a sector is \(A=\frac{1}{2}r^{2}\theta\), where \(\theta\) is the central angle in radians.
Substitute \(r = 2.8\), \(\theta=\frac{5\pi}{7}\), and \(\pi = 3.14\) into the formula.
First, \(r^{2}=(2.8)^{2}=7.84\).
Then, \(\frac{1}{2}r^{2}\theta=\frac{1}{2}\times7.84\times\frac{5\times3.14}{7}\).
\(\frac{1}{2}\times7.84 = 3.92\).
\(3.92\times\frac{5\times3.14}{7}=3.92\times\frac{15.7}{7}\).
\(3.92\div7 = 0.56\).
\(0.56\times15.7=8.792\).

Answer:

\(8.79\)