Problem 3: Two rectangular gardens are made from 200 yards of fencing. The
gardens are planted beside each other so they share a common fence. If the
farmer wants the total area of the gardens to be as large as possible, what
dimensions should be used to make the gardens, and what will the largest area
possible be?

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MrRoyal MrRoyal · Answer 1 · 2022-03-02T10:34:59+01:00

The area of a shape is the amount of space on the shape

The largest possible area of the garden is 1666.67 square yards

Represent the length of the gardens with y, and the widths of the whole garden with x.

So, the perimeter of the garden is:

[tex]P = 3y + 2x[/tex]

The farmer has 200 yards.

So, we have:

[tex]3y + 2x= 200[/tex]

The area of the garden is:

[tex]Area = xy[/tex]

Divide the amount of fencing equally to maximize the area.

So, we have:

[tex]3y + 2x = 100 + 100[/tex]

This gives

[tex]3y = 100 \to y = \frac{100}3[/tex]

[tex]2x = 100 \to x = 50[/tex]

Recall that:

[tex]Area = xy[/tex]

So, we have:

[tex]Area = 50 * \frac{100}3[/tex]

Evaluate the product

[tex]Area = \frac{5000}3[/tex]

[tex]Area = 1666.67[/tex]

Hence, the largest possible area of the garden is 1666.67 square yards