Question

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

Answer 1

Answer:

$324 ft^2$

Step-by-step explanation:

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

$A=Lw$ (1)

where

L is the length

w is the width

The perimeter of the rectangle is given by

$p=2L+2w$

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

$p=72 ft$

So we have:

$72=2L+2w$

Which can be rewritten as

$w=36-L$

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

$A=L(36-L)=36L-L^2$

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

$\frac{dA}{dL}=0$

Calculating the derivative,

$\frac{dA}{dL}=\frac{d}{dL}(36L-L^2)=36-2L$

And requiring it to be zero, we find:

$36-2L=0\\L=\frac{36}{2}=18$

Which means also

$w=36-L=36-18=18$

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:

$A=18\cdot 18=324 ft^2$

So, this is the maximum area.