Question

A circle has an arc length of 5π in. The central angle for this arc measures π/3 radians. What is the area of the associated sector?

Answer 1

The area of the associated sector is
`(25)/(24)\pi \ in^(2)`

Explanation:

step 1

Find the radius of the circle

we know that

The circumference of a circle is equal to

`C=2\pi r`

we have

`C=5\pi\ in`

substitute and solve for r

`5\pi=2\pi r`

`r=2.5\ in`

step 2

Find the area of the circle

we know that

The area of the circle is equal to

`A=\pi r^(2)`

we have

`r=2.5\ in`

substitute

`A=\pi (2.5^(2))=6.25\pi\ in^(2)`

step 3

Find the area of the associated sector

we know that

`2\pi\ radians` subtends the complete circle of area
`6.25\pi\ in^(2)`

so

by proportion

Find the area of a sector with a central angle of
`\pi/3\ radians`

`(6.25\pi )/(2\pi) =(x)/(\pi/3)\\x=6.25*(\pi/3)/2\\ \\x=(25)/(24)\pi \ in^(2)`