Question

A sector of a circle whose radius is r and whose angle is theta has a fixed perimeter P. How do you find the values of r and theta so that the area of the sector is a maximum?

Answer 1

`r=P/4` and `theta = 2`

Explanation:

The perimeter of the sector is two radii and the arc cut off by `theta`. So, the perimeter is given by

`P = 2r+rtheta`

The area of a sector is `A = 1/2r^2theta`

Since `P` is fixed, the only variables in `P = 2r+rtheta` are `r` and `theta`

`r = P/(2+theta)` and `theta = (P-2r)/r = P/r-2`. Is we rewrite for `A` using only `theta` we'll need the quotient rule to differentiate. So let's rewrite `A` using only `r`.

`A = 1/2r^2(P/r-2) = (Pr)/2 - r^2`

We want to maximize `A`, so . . .

`A' = P/2-2r = 0` at `r=P/4`

Note that `A'' = -2` so `A(P/4)` is a maximum, not a minimum.

Use `theta = P/r-2` from above to get `theta = 2`