Question

a sector of a circle with radius 15 ft has area 60π ft². what is the arc length of that sector? give your answer in terms of π or as a decimal rounded to two decimal places.
show your work here
hint to add the pi symbol (π), type \pi\

Explanation:

Step1: Recall the formula for the area of a sector

The area of a sector of a circle is given by the formula \( A=\frac{1}{2}r^{2}\theta \) (where \( \theta \) is in radians) or \( A=\frac{\theta}{360}\times\pi r^{2} \) (where \( \theta \) is in degrees), and the arc length of a sector is given by \( L = r\theta \) (when \( \theta \) is in radians) or \( L=\frac{\theta}{360}\times2\pi r \) (when \( \theta \) is in degrees). Also, we know the area of the whole circle is \( A_{circle}=\pi r^{2} \). The ratio of the area of the sector to the area of the circle is equal to the ratio of the arc length of the sector to the circumference of the circle. Let's use the formula \( \frac{A_{sector}}{A_{circle}}=\frac{L_{arc}}{C_{circle}} \), where \( A_{circle}=\pi r^{2} \), \( C_{circle} = 2\pi r \).

Given \( r = 15\) ft, \( A_{sector}=60\pi\) square feet. First, find the area of the circle: \( A_{circle}=\pi\times(15)^{2}=225\pi \) square feet.

Step2: Find the ratio of sector area to circle area

The ratio \( \frac{A_{sector}}{A_{circle}}=\frac{60\pi}{225\pi}=\frac{60}{225}=\frac{4}{15} \).

Step3: Relate the ratio to arc length

Since \( \frac{A_{sector}}{A_{circle}}=\frac{L_{arc}}{C_{circle}} \), and \( C_{circle}=2\pi r=2\pi\times15 = 30\pi \) feet. Let \( L \) be the arc length. Then \( \frac{4}{15}=\frac{L}{30\pi} \).

Step4: Solve for arc length \( L \)

Cross - multiply: \( 15L=4\times30\pi \)
\( 15L = 120\pi \)
Divide both sides by 15: \( L=\frac{120\pi}{15}=8\pi \)
If we want to write it as a decimal, \( 8\pi\approx8\times3.1415926535 = 25.1327412287\approx25.13 \) (rounded to two decimal places).

Answer:

If in terms of \( \pi \), the arc length is \( 8\pi \) feet. If as a decimal rounded to two decimal places, the arc length is \( 25.13 \) feet.