Question

A farmer wants to fence a rectangular area adjacent to a straight river with an extra partition running perpendicular to the river. If he has 3000 yards of fencing and no fencing is needed alone the river, then what is the maximum area he can enclose

Answer 1

the maximum area enclosed = 1125000sq. ft.

Explanation:

Let each side perpendicular to the river be "x".

Then the side parallel to the river is "3000-2x".

Now, area of the rectangle

A(x) = (3000-2x)x

A(x) = -2x^2+3000x

Which is a quadratic equation with a = -2 and b = 3000

Maximum area occurs where x = -b/2a

x = -3000/(-2×2) = 750 ft.

Therefore, width of rectangle x = 750 ft

Now, length = 3000-2(750) = 1500 ft.

Therefore, the maximum area enclosed = 1500×750 = 1125000 sq. ft.