Question

the angle $\theta$ in a circle with radius 8 in has an arc length of $6\pi$ in. what is the sector area with the same angle $\theta$? give your answer in terms of $\pi$ or as a decimal rounded to two decimal places.
show your work here
hint to add the pi symbol ($\pi$), type \pi\.

Explanation:

Step1: Recall arc length formula

The formula for arc length \( s \) is \( s = r\theta \), where \( r \) is the radius and \( \theta \) is the central angle in radians. We know \( s = 6\pi \) and \( r = 8 \), so we can solve for \( \theta \):
\( 6\pi = 8\theta \)
\( \theta = \frac{6\pi}{8} = \frac{3\pi}{4} \) radians.

Step2: Recall sector area formula

The formula for the area \( A \) of a sector is \( A=\frac{1}{2}r^{2}\theta \). We know \( r = 8 \) and \( \theta=\frac{3\pi}{4} \), substitute these values:
\( A=\frac{1}{2}\times8^{2}\times\frac{3\pi}{4} \)
First, calculate \( 8^{2}=64 \), then:
\( A=\frac{1}{2}\times64\times\frac{3\pi}{4} \)
Simplify \( \frac{1}{2}\times64 = 32 \), then \( 32\times\frac{3\pi}{4}=24\pi \).
If we want the decimal approximation, \( \pi\approx3.1416 \), so \( 24\pi\approx24\times3.1416 = 75.3984\approx75.40 \) (rounded to two decimal places).

Answer:

If in terms of \( \pi \), the area is \( 24\pi \) square inches. If as a decimal, the area is approximately \( 75.40 \) square inches.