Question

A sector shaped like a slice of pie is cut from a radius r the outer arc of the sector length s the total perimeter of the sector is 2r+s = 100m find the maximum value of the area of sector that is possible with this perimeter?

Answer 1

so if x is an angle of the sector and perimeter p=(2r+s), we will have

2r+s=100

s=2r*Pi*x/360=r*Pi*x/180=>

2r+r*Pi*x/180=100

area of the sector A=r^2*Pi*x*360

from the first equation x=(100-2r)*180/(r*Pi)

replace it in the area and we get area function as:

A=(100-2*r)*r/2= -r^2+50r

To find maximum solve dA/dr=0, which leads us to

-2*r+50=0 => r=25 to maximize the area