Question

Please Help? 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

Answer:

The area of the associated sector is $\frac{25}{24}\pi \ in^{2}$  

Step-by-step 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 $\pi/3\ radians$

$\frac{6.25\pi }{2\pi} =\frac{x}{\pi/3}\\x=6.25*(\pi/3)/2\\ \\x=\frac{25}{24}\pi \ in^{2}$