A farmer wants to build a rectangular pen which will be bounded on one side by a

Question

A farmer wants to build a rectangular pen which will be bounded on one side by a river and on the other three sides by a single-strand electric fence. If the farmer has 24 meters of wire to use, what is the largest area that the farmer can enclose? (You can click on the graphic to enlarge the image.) (a) Write the formula for the area A() of the pen as a function of r only (Refer to the given picture). A(x) square meters (b) The function A (x) has a critical point at meters (c) What is the maximum area of the pen? Answer: square meters. Important: On quizzes / exams you will be expected to use the techniques of MTH 132 to justify why your answer maximizes area.

Explanation / Answer

Area = x*y

2x+y = 24

y= 24 -2x

(a) A(x) = x*(24-2x)= 24x - 2x2 ..... (1)

(b) On differentiating equation (1) with x, we get

24 - 4x = 0

=> x = 6

So critical point is x = 6 meters

(c)

On double differentiating, we get -4 which is negative. So at x = 6 area is maximum.

The maximum area of the pen is = 24*6 - 2*6*6 = 36*(4-2) = 72 square meters

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