Question

Chris wants to make an enclosed rectangular area for a mulch pile. She wants to make the enclosure in such a way as to use a corner of her back yard. She also wants it to be twice as long as it is wide. Since the yard is already fenced, she simply needs to construct two sides of the mulch pile enclosure. She has only 15 feet of material available. Find the dimensions of the enclosure that will produce the maximum area

Answer 1

The dimensions for the mulch pile enclosure that will produce the maximum area with 15 feet of material are 10 feet in length and 5 feet in width, resulting in an enclosed area of 50 square feet.

Step-by-step explanation:

Chris wants to make the most of the 15 feet of material to create an enclosed rectangular area for a mulch pile where one side of the rectangle will be twice as long as the other. Because the area is already fenced, she only needs to construct two sides. The formula for the perimeter of a rectangle is P = 2l + 2w, but because the yard provides two sides, we adjust this to P = l + w, where l is the length and w is the width.

Since the length (l) is to be twice the width (w), we can express the length as l = 2w. Substituting the value of length into the perimeter equation we get 15 = 2w + w, which simplifies to 15 = 3w. Dividing both sides by 3 gives us w = 5 feet and, consequently, l = 10 feet.

The maximum area that can be enclosed with these dimensions is found by multiplying the length by the width: Area = l * w, which results in Area = 10 ft * 5 ft = 50 square feet.

Answer 2

The enclosure having the maximum area would be square. Since Chris wants the aspect ratio to be 2:1, she will use 2/3 of the fence (10 ft) for the long side and 1/3 of the fence (5 ft) for the short side.

The enclosure having the maximum area meeting Chris's requirement is 10 ft × 5 ft.