Answer to Question #168986 in Calculus for Shein

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 one side of rectangle will be x, and othe y. So it area will be S=xyTo find maximum value of x and y we should find derivatives of S_x and S_y.

S_x=y

S_y=x

So x=y

Maximum area of the lot will be, when it sides is equal. As along one side we dont need a fence, so to maximize the area we need to build one side of fence along the front and two sides along the sides. Let number of foots of the fence side will be x. So the front fence will be cost 75x, and the side fence will be cost 50x. As farmer wants to have a 24-ft opening, front fence will be 75(x-24) foots along.

75(x-24)+50x+50x=15000

175x=16800

x=96 foots

Answer: It will be square lot with side equal 96 foots