You want to put in a new pool in your yard and you decide to use a parallelogram shaped pool

Question

You have a maximum perimeter of 100 feet for the pool and you want the pool to have the maximum area.

What should the dimensions of the pool be to maximize the area?

Answer 1

In regards with the angle. You need to consider the area as a function of 2 variables:

Area(y, θ) = y*(y-50)*sinθ

To find the extremum, you need to solve two equations:

partial derivative by y is 0: sinθ(50-2y)=0

partial derivative by θ is 0: (50-y)*y*cosθ = 0

Which gives y=25 (as in solution above) and θ=90^º So it is indeed a square with the side = 25

Answer 2

If it's a rectangle, the we have area A = x*y

and perimeter: 2x + 2y = 100

Solve this for x to get x = 50 - y

Use this to replace y in the first equation:

A = (50 - y) * y = 50y - y^2

For the maximum A, take the 1st derivative and set = to 0:

A' = 50 - 2y = 0

y = 25

So, the maximum A for a rectangle comes when it is a square...

With a parallelogram, there is another variable--the angles. I think you will find that the maximum is still 4 equal sides and when the angles are all equal---i.e. when the parallelogram is a SQUARE!!