A farmer has 260 feet of fencing to make a rectangular corral. What dimensions will make a corral with the maximum area? What is the maximum area possible?
Thanks.
a square field has maximum area
To find the dimensions that will maximize the corral's area, we can use a bit of algebra and optimization principles.
Let's assume the length of the corral is L and the width is W. We know that the perimeter of a rectangle is given by the formula P = 2L + 2W. In this case, the perimeter is the total amount of fencing available, which is 260 feet. Therefore, we have the equation 2L + 2W = 260.
To find the dimensions that maximize the corral's area, we need to express the area in terms of a single variable and then use calculus to find the maximum point.
The area of a rectangle is given by the formula A = L * W. To express the area in terms of a single variable, we can solve the perimeter equation for L, which gives us L = (260 - 2W) / 2.
Substituting this value of L into the area equation, we get A = [(260 - 2W) / 2] * W.
Simplifying, we have A = 130W - W^2.
To find the maximum area, we can calculate the derivative of the area with respect to W, set it equal to zero, and solve for W.
dA/dW = 130 - 2W.
Setting this derivative equal to zero gives us the equation 130 - 2W = 0.
Solving for W, we find that W = 65.
We can then substitute this value of W back into the perimeter equation to find the corresponding value of L. Using 2L + 2W = 260, we have 2L + 2 * 65 = 260. Solving for L, we get L = 65.
Therefore, the dimensions that will make a corral with the maximum area are 65 feet by 65 feet. To find the maximum area, we can substitute these dimensions into the area equation. Using A = L * W, we get A = 65 * 65 = 4225 square feet.
So, the maximum area possible for the corral is 4225 square feet.