Question

How do you find the maximum area of a rectangle with perimeter of 40 ft?

Answer 1

Your gut feeling may tell you it's a square, with sides 10 ft, or a total area of 100 sqft.

Explanation:

You can divert from this and see if the area gets any larger, or you can use the mathematical way:
If the length =`x` and the width =`y` then the perimeter `P=40`
`P=2x+2y=40->x+y=20->y=20-x`

As for the area `A`:
`A=x*y=x*(20-x)=20x-x^2`
And we have to find an extreme for that:
We can do this by setting the derivative to `=0`
`A'=20-2x=0->x=10->y=10`
Just as we thought in the first place.
graph{20x-x^2 [-131.6, 135.4, -8.4, 125.1]}