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?

1 Answer
Jim H
Dec 4, 2016

#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#

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