A wedge-shaped solar panel is constructed in the shape of a circular sector with a central… — ACT

A wedge-shaped solar panel is constructed in the shape of a circular sector with a central angle measuring $\frac{3\pi}{4}$ radians. If the outer arc length of the panel is $9\pi$ feet, what is the area of the solar panel, in square feet?

A
$27\pi$
$54\pi$ Answer
C
$81\pi$
D
$108\pi$
E
$216\pi$

Answer

54\pi

square feet

First, determine the radius

r

of the sector using the arc length formula

s = r\theta

. Substituting

s = 9\pi

and

\theta = \frac{3\pi}{4}

gives

9\pi = r\left(\frac{3\pi}{4}\right)

, which simplifies to

r = 12

feet. Next, calculate the area of the sector using

A = \frac{1}{2}r^2\theta

. Substituting

r = 12

and

\theta = \frac{3\pi}{4}

yields

A = \frac{1}{2}(12)^2\left(\frac{3\pi}{4}\right) = 54\pi

square feet.

Step-by-Step Solution

Calculate the radius

r

of the circular sector using the arc length formula

s = r\theta

9\pi = r \left(\frac{3\pi}{4}\right) \implies r = 9\pi \cdot \frac{4}{3\pi} = 12\text{ feet}

The arc length

s

of a circular sector with central angle

\theta

in radians is

s = r\theta

Calculate the sector area using

A = \frac{1}{2}r^2\theta

with

r = 12

and

\theta = \frac{3\pi}{4}

A = \frac{1}{2}(12)^2\left(\frac{3\pi}{4}\right) = \frac{1}{2}(144)\left(\frac{3\pi}{4}\right) = 72 \cdot \frac{3\pi}{4} = 54\pi\text{ square feet}

The area of a sector with radius

r

and central angle

\theta

in radians is given by

A = \frac{1}{2}r^2\theta

Key Concept

Arc Length and Sector Area in Radians

Question

Answer

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