Optimizing rectangular garden. What is the minimum amount of fencing needed to enclose 2500m^2?

Question

a rectangular garden is split into 4 parts by constructing 3 additional fences parallel to one of the sides in addition to fencing the perimeter.

What is the minimum amount of fencing needed to enclose 2500 m^2?

Answer 1

First, as with most problems, start by drawing a good diagram of the situation to help picture and parameterize the situation. Unfortunately this text box doesn't give me a picture tool, so I'll draw it in ascii art, with fence pieces as X's, and unfenced area _. The rectangular area has sides length L and

H.

XXXXXXXXXXXXXXXXXXXXXXXXXXXX

X_____X______X________X___________X

X_____X______X________X___________X H

X_____X______X________X___________X

XXXXXXXXXXXXXXXXXXXXXXXXXXXX

L

So we have area LH, which we constrain to be the area (A). So LH = A = 2500m2. Then the perimeter (P) is 5H+2L, which you can from just counting fence panels in the diagram.

So we first isolate one of the unknowns from the area equation: L = A/H. Then put that into the perimeter equation:

P = 5H + 2A/H.

Now to find a minimum, we take a derivative dP/dH and set that equal to zero.