Dimensions of a Rectangle

Question

The rectangular bird sanctuary with one side along a straight river is to be constructed so that it contains 32 km²

of area. Find the dimensions of the rectangle to minimize the amount of fence necessary to enclose the remaining three sides.

Answer 1

Area A = L*w = 32 ---> L = 32/w

Perimeter P = 2L + 2W

= 2 (32/w) + 2w

= 64/w + 2w

= (64 +2w^2)/w

0= dP/dw = {w ( 4w) - (64 +2w^2) }/w^2

0 = 4w^2 - 64 - 2w^2

0 = 2w^2 - 64

w^2 = 32

w = 4*Sqrt(2)

L = 32/(4*sqrt(2)) = 8/sqrt(2) = 8*sqrt(2)/2 = 4*Sqrt(2)

Max perimeter is 16*sqrt(2)