Question

the radius of a quarter circle is 8 miles. what is the quarter circles area? round your answer to the nearest hundredth. square miles

Explanation:

Step1: Recall the area formula for a circle

The area of a full circle is given by the formula \( A = \pi r^2 \), where \( r \) is the radius of the circle.

Step2: Determine the area of the quarter circle

Since a quarter circle is one - fourth of a full circle, the area of a quarter circle \( A_{quarter} \) is \( \frac{1}{4} \) of the area of the full circle. So the formula for the area of a quarter circle is \( A_{quarter}=\frac{1}{4}\pi r^2 \).
We are given that the radius \( r = 8 \) miles. Substitute \( r = 8 \) into the formula:
\( A_{quarter}=\frac{1}{4}\times\pi\times(8)^2 \)
First, calculate \( 8^2=64 \). Then, \( \frac{1}{4}\times\pi\times64 = 16\pi \).
Using the approximation \( \pi\approx3.14159 \), we get \( 16\times3.14159 = 50.26544 \).

Step3: Round to the nearest hundredth

Rounding \( 50.26544 \) to the nearest hundredth (two decimal places), we look at the third decimal place which is 5. When the digit in the third decimal place is 5 or greater, we round up the second decimal place. So \( 50.26544\approx50.27 \).

Answer:

The area of the quarter circle is approximately \( 50.27 \) square miles.