SOLUTION: 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,
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Question 1205867: 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. Answer by ikleyn(52781) (Show Source):
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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.
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Imagine that three walls of fencing are perpendicular to the river (the length x yards)
and one wall is parallel to the river (the length y yards).
The total fence length equation is
3x + y = 480 yards. (1)
The area equation is area = x*y.
We want maximize the area under restriction (1).
From (1), express y = 480-3x and substitute into the expression for area.
You will get
area = x*(480-3x) = -3x^2 + 480x.
Thus you want to maximize this quadratic function.
The maximum of a quadratic function ax^2 + bx is achieved at vertex
= = = = 80 yards.
So, the three perpendicular walls are 80 yards each.
The wall parallel to the river is y = 480 - 3*80 = 240 yards.
The total area is 80*240 = 19200 sq. yards.
