Question

3) Puan wishes to build a garden at the bottom of his property. He wants to split in

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

Read more about maximum areas at:

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