Question

A rope of length 18 feet is arranged in the shape of a sector of a circle with central angle O radians, as shown in the

accompanying figure. Write the area of the sector. A as a function of ​

Answer 1

`A(\theta)=(162 \theta)/((\theta+2)^2)`

Explanation:

The picture of the question in the attached figure

step 1

Let

r ---> the radius of the sector

s ---> the arc length of sector

Find the radius r

we know that

`2r+s=18`

`s=r \theta`

`2r+r \theta=18`

solve for r

`r=(18)/(2+\theta)`

step 2

Find the value of s

`s=r \theta`

substitute the value of r

`s=(18)/(2+\theta)\theta`

step 3

we know that

The area of complete circle is equal to

`A=\pi r^(2)`

The complete circle subtends a central angle of 2π radians

so

using proportion find the area of the sector by a central angle of angle theta

Let

A ---> the area of sector with central angle theta

`(\pi r^(2) )/(2\pi)=(A)/(\theta) \\\\A=(r^2\theta)/(2)`

substitute the value of r

`A=(((18)/(2+\theta))^2\theta)/(2)`

`A=(162 \theta)/((\theta+2)^2)`

Convert to function notation

`A(\theta)=(162 \theta)/((\theta+2)^2)`