Question

A contractor is to fence off a rectangular field along a straight river, the side along the river requiring no fence.
What is the least amount of fencing needed to fence of 30,000 meters squared?

Answer 1

perimeter = L + 2 W = p

L W = 30,000
so
L = 30,000/W
then
p = 30,000/W + 2 W

dp/dW = -30,000/W^2 + 2
for in or max, dp/dW = 0
30,000 = 2 W^2
W^2 = 15,000
W = 122.5
then L = 30,000/122.5 = 244.9

Answer 2

above gives p = 244.9+2*122.5 = 490

If you do not know derivatives then

p = 30,000/W + 2 W
2 W^2 - Wp +30,000 = 0
W^2 - (p/2)W = -15,000
W^2 - (p/2)W + p^2/16 = -15,000 + p^2/16

(W-p/4)(W-p/4) = 1/16( p^2 - 240,000)
that is vertex at W = p/4 and p = sqrt(240,000) or 490 sure enough