If you were given a rectangle with a certain area how would you draw it so that it had the greatest perimeter

Anwer: draw a square with side length equal to the square root of the area of the rectangle.

Explanation:

The rectangle that has the greatest perimeter given a fixed area is the square.

So, take the square root of the area and draw a square with that side length.

The demostration of that is done using the optimization concept from derivative. If you already studied derivatives you can follow the following demostration.

These are the steps:

1) dimensions of the rectangle:

length: l
width: w

perimeter formula: p = 2l + 2w
area formula: A = lw

2) solve l or w from the area formula: l = A / w

3) write the perimeter as a function of w:

p = 2 (A / w) + 2w

4) find the derivative of the perimeter, dp / dw = p'

p' = - 2A / w^2 + 2

5) The condition for optimization is p' = 0

=> -2A / w^2 + 2 = 0

=> 2A / w^2 = 2

=> w^2 = A

Which means that the dimensions of the rectangle are w*w, i.e. it is a rectangle of side length w = √A