Find the length and width of a rectangle that has the given perimeter and a maximum area. Perimeter: 9P units

Question

This is an optimization problem. So I think I have to find the constraint function and the object function, but I am at a loss.

Answer 1

Let L and W represent length and width of rectangle respectively. Then 2L + 2W = 9P.

We know A = LW, so find and expression for L in terms of W (or vice versa).

2L = 9P - 2W

L = (9/2)P - W.

So area in terms of W only is:

A = [(9.2)P - W]W = (9/2)PW - W².

Since this question was posed under the topic Calculus we can find the derivative of A with respect to W (remember that P is a constant).

dA/dW = (9/2)P -2W

Setting equal to zero to find critical number(s) results in:

2W = (9/2)P

W = (9/4)P

Notice that the 2nd derivative is always negative, so the function's graph is always concave downward and the value (9/4)P generates a maximum value for the area function.

L = (9/2)P - W= (9/2)P - (9/4)P = (9/4)P , so L = W and the maximum area happens when the rectangle is a square.

A = (9/4)P (9/4)P = (81/16)P²

Here is a Desmos graph that validates:

desmos.com/calculator/91oo14wrqk

Notice that regardless of the value for P greater than zero, the vertex of the area function matches the above.