Questions: A rancher needs to enclose two adjacent rectangular corrals, one for cattle and one for sheep. If the river forms one side of the corrals and 480 yd of fencing is available, find the largest total area that can be enclosed. What is the largest total area that can be enclosed? square yd^2

Solution Steps

Let \( x \) be the width of each corral and \( y \) be the length of the corrals.

The total fencing available is 480 yards. The river forms one side of the corrals, so the fencing is used for the other three sides and the internal divider: \[ 3x + 2y = 480 \]

The total area \( A \) of the two corrals is: \[ A = x \cdot y \]

From the perimeter constraint, solve for \( y \): \[ y = \frac{480 - 3x}{2} \]

Substitute \( y \) into the area formula: \[ A = x \left( \frac{480 - 3x}{2} \right) \] \[ A = \frac{480x - 3x^2}{2} \] \[ A = 240x - \frac{3x^2}{2} \]

To find the maximum area, take the derivative of \( A \) with respect to \( x \) and set it to zero: \[ \frac{dA}{dx} = 240 - 3x = 0 \] \[ 3x = 240 \] \[ x = 80 \]

Substitute \( x = 80 \) back into the equation for \( y \): \[ y = \frac{480 - 3(80)}{2} \] \[ y = \frac{480 - 240}{2} \] \[ y = 120 \]

Substitute \( x = 80 \) and \( y = 120 \) into the area formula: \[ A = 80 \cdot 120 \] \[ A = 9600 \]

Final Answer