A homeowner wants to enclose a rectangular garden with fencing The garden will be adjacent to his neighbours lot. There will be fencing on all four sides. His neighbour will be paying for half of the shared fence.

Question

What should the dimensions of the garden be if the area is 432 m^2 and the homeowner would like to keep his share of the cost to a minimum.

Good day, I'm currently stuck on my last homework question and have failed three times to answer it properly... I have worked with questions like this before but I was asked to maximize the area when given the total length of fencing used but I'm confused as to where to start when given the area...

Thank you very much for your time, it's very appreciated.

Steve · Answer

If the dimensions are x and y, then
xy = 432, so y = 432/x

If the shared side is y, then the cost for our homeowner is

c = 2x + y + y/2 = 2x + 3/2 y = 2x + 648/x

we want minimum cost, so

dc/dx = 2 - 648/x^2
dc/dx = 0 when x = 18

So, the garden is 18x24, with the 24' side shared.

Scott · Answer

Thank you very much Steve.