Question

the angle \\( \theta \\) in a circle with radius 11 ft has an arc length of \\( 3\pi \\) ft. 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.

Explanation:

Step1: Recall the arc length formula

The arc length formula is \( s = r\theta \), where \( s \) is the arc length, \( r \) is the radius, and \( \theta \) is the central angle in radians. We know \( s = 3\pi \) and \( r = 11 \), so we can find \( \theta \) first, but actually, for the sector area, we can also use the formula \( A=\frac{1}{2}rs \) (since \( A = \frac{1}{2}r^{2}\theta \) and \( s=r\theta \), so substituting \( s \) into the area formula gives \( A=\frac{1}{2}rs \)).

Step2: Apply the sector area formula

We are given \( r = 11 \) ft and \( s = 3\pi \) ft. Using the formula \( A=\frac{1}{2}rs \), we substitute the values:
\( A=\frac{1}{2}\times11\times3\pi \)

Step3: Calculate the value

First, calculate \( \frac{1}{2}\times11\times3=\frac{33}{2} = 16.5 \), so \( A = 16.5\pi \). If we want to write it as a decimal, \( \pi\approx3.1416 \), so \( A = 16.5\times3.1416\approx51.84 \) (rounded to two decimal places). But let's check with the formula \( A=\frac{1}{2}r^{2}\theta \) as well. From \( s = r\theta \), \( \theta=\frac{s}{r}=\frac{3\pi}{11} \). Then \( A=\frac{1}{2}\times11^{2}\times\frac{3\pi}{11}=\frac{1}{2}\times11\times3\pi \), which is the same as before. So \( A=\frac{33\pi}{2}=16.5\pi\approx51.84 \) square feet.

Answer:

The sector area is \( \frac{33\pi}{2} \) square feet or approximately \( 51.84 \) square feet. If we use the exact form in terms of \( \pi \), it's \( 16.5\pi \) (or \( \frac{33\pi}{2} \)), and as a decimal, it's approximately \( 51.84 \) square feet.