The biggest pen you can build against a wall

Question

A farmer has 100 meters of fencing and wants to build a rectangular animal pen. Luckily, one side of the pen can run along an existing long straight wall, so no fence is needed there, the fence only needs to cover the other three sides. What dimensions give the largest possible enclosed area, and how much bigger is it than fencing all four sides?

Accepted Answer

Make the pen 50 m (along the wall) by 25 m (deep), enclosing 1250 square meters. Set up the area. Let the two sides perpendicular to the wall each have length x; then the side parallel to the wall uses the leftover fence, 100 - 2x. The enclosed area is A = x \,(100 - 2x) = 100x - 2x^2. Maximize it. This is a downward parabola in x, largest at its vertex. The vertex of 100x - 2x^2 sits at x = \dfrac{100}{2 	imes 2} = 25. (Equivalently, the two "x" sides should together use half the fence.) So: - Depth: x = 25 m (each of the two perpendicular sides). - Width along the wall: 100 - 2(25) = 50 m.…

The biggest pen you can build against a wall

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