Question

3) Puan wishes to build a garden at the bottom of his property. He wants to split in
two, like the diagram below, so he can plant vegetables in one side and fruit in the
other. There is a stream at the bottom of his property so he doesn't have to fence
there. He has 120 feet of fencing. Find the function that models this scenario (4)
(3)
a) What should the dimensions of the garden be to maximize this area?
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(2
b) What is the maximum area that Juan can enclose for his garden?

Answer 1

The area of the garden is the product of its dimensions

• The dimension that maximizes the area is 20 by 60
• The maximum area is 1200 square feet

From the figure (see attachment), the perimeter is:

$\mathbf{P = 3x + y}$

The perimeter is given as: 120.

So, we have:

$\mathbf{3x + y = 120}$

Make y the subject

$\mathbf{y = 120 - 3x}$

The area is then calculated as:

$\mathbf{A =xy}$

Substitute $\mathbf{y = 120 - 3x}$

$\mathbf{A =x(120 - 3x)}$

$\mathbf{A =120x- 3x^2}$

Differentiate

$\mathbf{A' =120- 6x}$

Set to 0

$\mathbf{120- 6x = 0}$

Rewrite as:

$\mathbf{6x = 120}$

Divide through by 6

$\mathbf{x = 20}$

Recall that: $\mathbf{y = 120 - 3x}$

So, we have:

$\mathbf{y = 120 - 3(20)}$

$\mathbf{y = 120 - 60}$

$\mathbf{y =60}$

The dimension that maximizes the area is 20 by 60

The area is calculated as:

$\mathbf{A =xy}$

So, we have:

$\mathbf{A = 20 \times60}$

$\mathbf{A = 1200}$

Hence, the maximum area is 1200 square feet

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