College Algebra

Question

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 360 yd of fencing is available, find the largest total area that can be enclose.

What is the largest total area that can be enclosed?

__yd^2

Answer 1

Since they are adjoining rectangular corrals, that would mean the fence of one side will be a fence for both corrals' adjoining side

That leaves 360 yards of fence for the side parallel to the river and 3 sides perpendicular to the river

L+3W = 360

or L=360-3W = 3(120-W)

If the areas of each are the same then

the area of each is (L/2)W = [3(120-W)/2]W = 180W - 3W^2/2

take the derivative with respect to W and set = 0 to get

-3W+180 = 0

solve for W

3W=180

W = 60

L=360-3(60) = 180

each corral is 60x180 = 10800 square yards

total for both = 2(10800) = 21600 square yards

But if you took all the fencing for one corral and zero for the other

the area of the large corral would be LW = 360-2W x W = 360W -2W^2

take the derivative and set = 0

360-4W = 0

4W =360

W = 360/4 = 90

L=360-2(90) = 180

Total area = 90x180 = 16200 square yards

Maximum area is making two corrals of equal size, dimensions and area 60x180 = 21,600 yd^2

Making one corral larger than the other reduces the total area