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

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
Transcript text: A rancher needs to enclose two adjacent rectangular cortals, one for cattle and one for sheep. If the river forms one side of the corrals and 480 yd of tencing is available, find the largest total area that can be enclosed. What is the largest total area that can be enclosed? $\square \mathrm{yd}^{2}$
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Solution

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Solution Steps

Step 1: Define Variables

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

Step 2: Express the Perimeter Constraint

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 3x + 2y = 480

Step 3: Express the Area to Maximize

The total area A A of the two corrals is: A=xy A = x \cdot y

Step 4: Solve for One Variable

From the perimeter constraint, solve for y y : y=4803x2 y = \frac{480 - 3x}{2}

Step 5: Substitute into the Area Formula

Substitute y y into the area formula: A=x(4803x2) A = x \left( \frac{480 - 3x}{2} \right) A=480x3x22 A = \frac{480x - 3x^2}{2} A=240x3x22 A = 240x - \frac{3x^2}{2}

Step 6: Find the Maximum Area

To find the maximum area, take the derivative of A A with respect to x x and set it to zero: dAdx=2403x=0 \frac{dA}{dx} = 240 - 3x = 0 3x=240 3x = 240 x=80 x = 80

Step 7: Calculate Corresponding y y

Substitute x=80 x = 80 back into the equation for y y : y=4803(80)2 y = \frac{480 - 3(80)}{2} y=4802402 y = \frac{480 - 240}{2} y=120 y = 120

Step 8: Calculate the Maximum Area

Substitute x=80 x = 80 and y=120 y = 120 into the area formula: A=80120 A = 80 \cdot 120 A=9600 A = 9600

Final Answer

The largest total area that can be enclosed is 9600 9600 square yards.

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