Question

a circular sector has an area of 27 in.². the radius of the circle is 3 in. what is the arc length of the sector? the arc length of the sector is □ in. (type an integer or a decimal.)

Explanation:

Step1: Recall the formula for the area of a sector

The area of a sector \( A \) is given by \( A=\frac{1}{2}r^{2}\theta \), where \( r \) is the radius and \( \theta \) is the central angle in radians. We know \( A = 27\) \( \text{in}^2\) and \( r=3\) in. Substitute these values into the formula:
\( 27=\frac{1}{2}\times(3)^{2}\times\theta \)

Step2: Solve for \( \theta \)

First, simplify the right - hand side: \( \frac{1}{2}\times9\times\theta=\frac{9}{2}\theta \)
So the equation becomes \( 27 = \frac{9}{2}\theta \). Multiply both sides by \( \frac{2}{9} \) to solve for \( \theta \):
\( \theta=27\times\frac{2}{9}=6 \) radians.

Step3: Recall the formula for arc length of a sector

The arc length \( s \) of a sector is given by \( s = r\theta \), where \( r \) is the radius and \( \theta \) is the central angle in radians. We know \( r = 3\) in and \( \theta=6 \) radians.

Step4: Calculate the arc length

Substitute \( r = 3\) and \( \theta = 6\) into the arc - length formula: \( s=3\times6 = 18 \) in.

Answer:

18