Question

A farmer decides to make three identical pens with 104 feet of fence. The pens will be next to each other sharing a fence and will be up against a barn. The barn side needs no fence. What dimensions for the total enclosure (rectangle including all pens) will make the area as large as possible?

Answer 1

To maximize the area of the total enclosure, we need to find the dimensions that will make the area as large as possible.

Step-by-step explanation:

To find the dimensions of the total enclosure that will make the area as large as possible, we need to maximize the area of the rectangle. Let's assume the width of each pen is 'x', then the length of each pen will be (104 - 2x)/3. The dimensions of the rectangle including all pens will be 3 times the width of a single pen and the length will be the same as the length of a single pen.

So, the area of the rectangle is given by:

A = 3x * (104 - 2x)/3

To maximize the area, we can take the derivative of the area function with respect to 'x' and set it equal to zero. Solving this equation will give us the critical points and we can determine which one maximizes the area. Once we find the value of 'x', we can substitute it back into the area equation to find the dimensions of the total enclosure.