Final answer:
To find the dimensions of a rectangle with the maximum area, we need to maximize its length and width while keeping the perimeter constant. We can express the area of the rectangle in terms of either the length or the width, depending on which one we choose to solve for first. To find the maximum area, we take the derivative of the area with respect to the chosen variable and set it equal to 0.
Step-by-step explanation:
To find the dimensions of a rectangle with the maximum area, we need to maximize its length and width while keeping the perimeter constant. In this case, the perimeter is given as 124 feet. Let's assume the length of the rectangle is L and the width is W.
Given that perimeter = 2(L + W) = 124 feet, we can rearrange the equation to solve for either L or W. Let's solve for L:
L = (124 - 2W) / 2
Now we can express the area of the rectangle in terms of W:
Area = L * W = ((124 - 2W) / 2) * W
To find the maximum area, we can take the derivative of the area with respect to W and set it equal to 0 to find the critical point:
d(Area)/dW = 0
Solving this equation will give us the value of W that maximizes the area.
Once we have the value of W, we can substitute it back into the equation for L to find the dimensions of the rectangle with the maximum area.