Question: 4) A rancher wants to construct two identical rectangular corrals using 400 ft of fencing. The…

4. A rancher wants to construct two identical rectangular corrals using 400 ft of fencing. The rancher decides to build them adjacent to each other, so they share fencing on one side. What dimensions should the rancher use to construct each corral so that together, they will enclose the largest possible area?

Solution

To solve the problem, we need to find the dimensions of the two identical rectangular corrals that maximize the total enclosed area. Let \( x \) be the width of each corral, and \( y \) be the length of each corral. Since the corrals share one side, the total fencing used is: \[ 3x + 2y = 400 \] We need to maximize the area \( A \) of the two corrals: \[ A = 2xy \] First, solve the perimeter constraint for \( y \): \[ 2y = 400 - 3x \] \[ y = \frac{400 - 3x}{2} \] Substitute \( y \) into the area equation: \[ A = 2x \left(\frac{400 - 3x}{2}\right) \] \[ A = x(400 - 3x) \] \[ A = 400x - 3x^2 \] This is a quadratic equation in the form of \( A = -3x^2 + 400x \). To find the maximum area, determine the vertex of the parabola. The maximum occurs at: \[ x = -\frac{b}{2a} = -\frac{400}{2(-3)} = \frac{400}{6} = \frac{200}{3} \] Substitute \( x = \frac{200}{3} \) back into the equation for \( y \): \[ y = \frac{400 - 3\left(\frac{200}{3}\right)}{2} = \frac{400 - 200}{2} = 100 \] Therefore, the dimensions that maximize the total area are: Width \( x = \frac{200}{3} \) feet Length \( y = 100 \) feet.