Answer:
[tex]A(\theta)=\frac{162 \theta}{(\theta+2)^2}[/tex]
Step-by-step 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
[tex]2r+s=18[/tex]
[tex]s=r \theta[/tex]
[tex]2r+r \theta=18[/tex]
solve for r
[tex]r=\frac{18}{2+\theta}[/tex]
step 2
Find the value of s
[tex]s=r \theta[/tex]
substitute the value of r
[tex]s=\frac{18}{2+\theta}\theta[/tex]
step 3
we know that
The area of complete circle is equal to
[tex]A=\pi r^{2}[/tex]
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
[tex]\frac{\pi r^{2} }{2\pi}=\frac{A}{\theta} \\\\A=\frac{r^2\theta}{2}[/tex]
substitute the value of r
[tex]A=\frac{(\frac{18}{2+\theta})^2\theta}{2}[/tex]
[tex]A=\frac{162 \theta}{(\theta+2)^2}[/tex]
Convert to function notation
[tex]A(\theta)=\frac{162 \theta}{(\theta+2)^2}[/tex]
