Question

Find the dimensions of a rectangle that give the maximum area of the
rectangle, given that the perimeter of the rectangle is 36m.
Tuskegee University

Answer 1

Perimeter of the rectangle

P=2(x+y)

36=2(x+y)

x+y=18

y=18-x ------------(1)

Area of the rectangle

A=xy

A=x(18-x) (putting the value of y from equation (1))

A=18x - x^2

differentiating with respect to x

=18-2x

for critical points

A'(x)=0

18-2x=0

2x=18

x=9

second derivative test

so x=9 m is maxima point

from equation (1)

y=18-9=9 m

So dimensions

x=9 m ; y=9m

Amax=9*9=81 m^2