We are given the description of an enclosure that has fencing on 3 sides and can use 3,056 yards of fencing. We are to find the dimensions that will maximize the area contained within the fence. A rectangle formed along a straight portion of a river. The rectangle is formed by the river (a long side), a long side of fence labeled y, and two short sides of fence labeled x. Let y represent the length of fence that is opposite the river and x represent the length of each length of fence perpendicular to the river. As there are 3,056 yards of fencing available, the sum of the lengths of the sides of the enclosure results in

The sum of the lengths of the sides is 2292 yards and the sum of the lengths of the triangle is 3056 yards

Explanation:

Since y represents the length of fence that is opposite (parallel) to the river and x represent the length of fence perpendicular to the river.

Therefore since we can use 3,056 yards of fencing

Side perpendicular to the river = x and,

Side opposite to the river = y = 3056 - 2x

The area of the rectangle formed (A) = Perpendicular side × Parallel side

∴ A = x(3056 - 2x) = 3056x - 2x²

A = 3056x - 2x²

To maximize the area, A' (dA/dx) = 0

∴ A' = 3056 - 4x = 0

3056 - 4x = 0

4x = 3056

x = 764 yards

y =  3056 - 2x = 3056 - 2(764) = 1528 yards.

Side perpendicular to the river = 764 yards and,

Side opposite to the river = 1528 yards

The sum of the lengths of the sides = 764 + 1528 = 2292 yard and the sum of the lengths of the triangle = 764 + 764 + 1528 = 3056 yards

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