You have 400 feet of fencing to construct a rectangular pen cattle.what are the dimensions of the pen that maximizes the area?

To maximize the area of the rectangular pen, we need to determine the dimensions that will allow us to use all of the available fencing (400 feet). Let's denote the length of the pen as \( l \) and the width as \( w \). Since the pen is rectangular, the perimeter (total length of fencing) is given by the formula: Perimeter = 2l + 2w We are given that the total length of fencing is 400 feet, so: 2l + 2w = 400 l + w = 200 l = 200 - w Now, the area (A) of the rectangular pen is given by the formula: A = l * w Substituting the expression for l obtained above: A = (200 - w) * w A = 200w - w^2 To find the dimensions that maximize the area, we take the derivative of the area with respect to w, set it equal to 0, and solve for w. Then we use this value of w to find the corresponding value of l. Let's find the critical points: dA/dw = 200 - 2w Setting this derivative equal to 0: 200 - 2w = 0 2w = 200 w = 100 So, when w = 100 feet, the area is maximized. Now, we can find the corresponding value of l: l = 200 - w l = 200 - 100 l = 100 Therefore, the dimensions of the rectangular pen that maximize the area are: Length (l) = 100 feet Width (w) = 100 feet