Question

48 feet of fencing is used to create a [-

pen enclosure which consists of three

congruent rectangular pens as shown,

with a combined area of 72 ft?. What

is the perimeter of the enclosure?

Answer 1

The perimeter of the entire enclosure, calculated using the derived values of W and L, is exactly 48 feet

How to determine perimeter?

The total area of the three pens is 72 ft², and since they are congruent, the area of one pen is
`\( (72)/(3) = 24 \) ft^2`.

Therefore, the area equation for one pen is
`\( L * W = 24 \) ft^2`.

The perimeter of the entire enclosure, considering the shared sides, is
`\( 3W + 4L = 48 \)` feet.

From the area equation, express L in terms of W:

`\( L = (24)/(W) \).`

Substitute
`\( L = (24)/(W) \)` into the perimeter equation:

`\( 3W + 4 * (24)/(W) = 48 \)`.

Solving this equation for W:

`\( W = 8 - 4√(2) \) feet.`

Substitute
`\( W = 8 - 4√(2) \)` into
`\( L = (24)/(W) \)` to find L.

This gives
`\( L \approx 10.24 \)` feet.

Finally, confirm the perimeter using these values:

`\( P = 3W + 4L \).`

Substituting
`\( W = 8 - 4√(2) \)` and
`\( L \approx 10.24 \)` into this equation:

`\( P = 48 \)` feet, which matches the given length of fencing.

Therefore, the dimensions found are consistent with the given total area and fencing length, and the perimeter of the enclosure is indeed 48 feet.