Question

A rectangular corral is enclosed using 120 m of fencing. The corral uses the side of a barn as one of its sides, so fencing is required on three sides of the rectangular region. What dimensions would maximize the area enclosed by the fence?

Answer 1

To maximize the area enclosed by the fence, we need to find the dimensions of the corral. By setting up an equation based on the perimeter and solving for the dimensions, we can find the maximum area. The dimensions that maximize the area enclosed by the fence are 30 m by 60 m.

Step-by-step explanation:

To find the dimensions that will maximize the area enclosed by the fence, we need to use the perimeter of the corral. Since the corral uses one side of the barn, there are 3 sides that need fencing. Let the length of one of the sides (AB) be x and the width of the corral (BC or CD) be y. From the given information, we know that the perimeter is 120 m. Therefore, the equation representing the perimeter is:

2x + y = 120

To find the dimensions that maximize the area, we need to solve for y in terms of x:

y = 120 - 2x

Now we can find the area of the corral in terms of x:

A = x * y = x * (120 - 2x)

To find the maximum area, we take the derivative of the area function and set it equal to zero:

A' = 120x - 2x^2 = 0

Solving this quadratic equation, we get x = 30 and x = 0 (which does not make sense in this context). Therefore, the length of one side of the corral is 30 m. Substituting this value back into the equation for y, we get y = 60. So the dimensions that maximize the area enclosed by the fence are 30 m by 60 m.

Answer 2

The dimensions would be 40 m by 40 m.

To maximize area and minimize perimeter we make the dimensions as close to equal as possible. Since we only need fencing on 3 sides, 120/3 = 40; we can use 40 for each side.