Answer to Question #169279 in Calculus for Nics

Question

A farmer plans to fence his rectangular lot to secure his plantation. The lot is bounded at the back by a river; hence, no fence is needed along this side. In the front, the farmer wants to have a 24-ft opening. He surveyed the cost of the fence and noted that the fence along the front costs Php 75 per ft and Php 50 per ft along the sides. His budget for the fence is Php 15,000. 

As an architect, you were asked to prepare a plan for the fence. The plan is expected to present the 

dimensions of the largest lot that can be fenced given the budget.

Answer 1

Let the length be x and width be y

Area = x * y

The constraint is

75(x - 24) + 2(50)(y) = 15000 => y = 168 - 0.75x

Area = x(168 - 0.75x)

We wish to maximize area

At max area, d(Area)/dx = 0

168 - 1.5x = 0

x = 112

y = 84

d²(Area)/dx² = -1.5, which is a maximum

Hence the dimensions that maximize the lot is 112 by 84 feet