A gardener is fencing off a rectangular area with a fixed perimeter of 72 ft. What is the maximum area?

Step-by-step explanation:

The area of a triangle is given by the product between length and width:

(1)

where

L is the length

w is the width

The perimeter of the rectangle is given by

In this problem, we know that the perimeter of the rectangle is fixed, and it is

So we have:

Which can be rewritten as

If we substitute this into the formula of the area, (1), we get:

To maximize the area, we have to calculate its derivative and require it to be equal to zero:

Calculating the derivative,

And requiring it to be zero, we find:

Which means also

So,

L = 18 feet

w = 18 feet

So the maximum area is achieved when the rectangle has actually the shape of a square.

In such case, the area is:

So, this is the maximum area.

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