Optimization Question

Question

A dairy farmer plans to fence in a rectangular pasture adjacent to a river. The farmer only has 1200 ft of fencing. What are the dimensions and the area of the largest rectangular pasture that can be constructed with 1200 ft of fencing? Assume that no fencing is needed along the river.

Answer 1

Let x = the shorter side of the fence (perpendicular to the river), then 1200-2x would be the longer side of the fence (parallel to the river).

Area = l x w = x(1200-2x) = 1220x - 2x^2 Take the first derivative and set equal to zero

1200 - 4x = 0

x = 300 (width) and 600=length