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
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?
$\square \mathrm{yd}^{2}$
Solution
Solution Steps
Step 1: Define Variables
Let x be the width of each corral and 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
Step 3: Express the Area to Maximize
The total area A of the two corrals is:
A=x⋅y
Step 4: Solve for One Variable
From the perimeter constraint, solve for y:
y=2480−3x
Step 5: Substitute into the Area Formula
Substitute y into the area formula:
A=x(2480−3x)A=2480x−3x2A=240x−23x2
Step 6: Find the Maximum Area
To find the maximum area, take the derivative of A with respect to x and set it to zero:
dxdA=240−3x=03x=240x=80
Step 7: Calculate Corresponding y
Substitute x=80 back into the equation for y:
y=2480−3(80)y=2480−240y=120
Step 8: Calculate the Maximum Area
Substitute x=80 and y=120 into the area formula:
A=80⋅120A=9600
Final Answer
The largest total area that can be enclosed is 9600 square yards.