Question

a sector of a circle with radius 14 cm has area 70π cm². 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 \( A = \frac{1}{2} r l \), where \( r \) is the radius of the circle and \( l \) is the arc length of the sector. We are given that \( r = 14 \) cm and \( A = 70\pi \) \( \text{cm}^2 \).

Step2: Substitute the known values into the formula

Substitute \( A = 70\pi \) and \( r = 14 \) into the formula \( A=\frac{1}{2}rl \):
\[
70\pi=\frac{1}{2}\times14\times l
\]

Step3: Simplify and solve for \( l \)

First, simplify the right - hand side: \( \frac{1}{2}\times14 = 7 \), so the equation becomes \( 70\pi=7l \).
Then, divide both sides of the equation by 7 to solve for \( l \):
\[
l=\frac{70\pi}{7}=10\pi
\]
If we want to write it as a decimal, we know that \( \pi\approx3.14159 \), so \( l = 10\times3.14159=31.4159\approx31.42 \) (rounded to two decimal places).

Answer:

If in terms of \( \pi \), the arc length is \( 10\pi \) cm. If as a decimal rounded to two decimal places, the arc length is \( 31.42 \) cm.