Answer :

Final answer:

To find the length and width of a rectangle with the largest possible area given a perimeter of 120ft, use the concept of maximizing the area by setting up an equation and finding the critical points of the area function.

Explanation:

To find the length and width of a rectangle with the largest possible area given a perimeter of 120ft, we need to use the concept of maximizing the area of a rectangle.

Let's assume the length of the rectangle is L and the width is W. Since the perimeter is given as 120ft, we can set up the equation 2L + 2W = 120. Simplifying the equation, we get L + W = 60.

To find the maximum area, we need to express the area in terms of one variable. Solving the previous equation for L, we get L = 60 - W. Substituting this into the formula for the area, A = L × W, we get A = (60 - W) × W = 60W - W^2.

To find the maximum area, we need to find the critical points of the area function. Taking the derivative of A with respect to W and setting it equal to zero, we get dA/dW = 60 - 2W = 0. Solving for W, we find that W = 30.

Substituting this value into the equation L + W = 60, we find that L = 30. Therefore, the rectangle with the largest possible area and a perimeter of 120ft has a length of 30ft and a width of 30ft.

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