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Liono4ka [1.6K]
3 years ago
10

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

Mathematics
1 answer:
Anna71 [15]3 years ago
8 0

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.

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