Optimization word problem

Question

A farmer plans to enclose a rectangular pasture adjacent to a river. (see figure). The pasture must contain 320,000 square meters in order to provide enough grass for the herd. What dimensions will require the least amount of fencing if no fencing is needed along the river?

y = ? (m)

x = ? (m)

any help will be greatly appreciated, thank you in advance!

Answer 1

320,000 square meters = Area = LW = length times width

L=320,000/W

Fencing = Perimeter = P = L+2W= 320,000/W +2W

take the derivative and set = 0

P'(W) = 2 -320,000/W^2 = 0

2W^2 = 320,000

W^2 = 160,000

W = sqr160,000= 400 meters wide

L =320,000/400 = 800 meters long, the side parallel to the river

Answer 2

Draw a diagram.

If y is the length and x is the width then xy = 320000, and we want to minimise length of fence

L = 2x+y

xy = 320000, so y = 320000 / x

L = 2x + y = 2x + (320000/x)

So plot y = 2x + (320000/x) on Desmos and choose x giving the smallest y.

I get x = 400, so y = 320000/x = 800 so fence is 800 long (parallel to the river) and the other two sides are 400.