Consider the following problem: A farmer with 750 ft of fencing wants to enclose
ID: 2893619 • Letter: C
Question
Consider the following problem: A farmer with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens? (a) Drawseveraldiagramsillustratingthesituation,some with shallow, wide pens and some with deep, narrow pens. Find the total areas of these con gurations. Does it appear that there is a maximum area? If so, estimate it. (b) Draw a diagram illustrating the general situation. Intro- duce notation and label the diagram with your symbols. (c) Write an expression for the total area. (d) Use the given information to write an equation that relates the variables. (e) Use part (d) to write the total area as a function of one variable. (f) Finish solving the problem and compare the answer with your estimate in part (a).
Explanation / Answer
(a) Here are three diagrams. The area of the first one is 12500 ft2 with 50 ft height and 250 ft width , the area of the second one is 12500 ft2 with 100 ft height and 125 ft width , and the area of the third one is 9000 ft2 with 75 ft width and 120 ft height
It seems that there is a maximum area and I would put it at around 13000 ft2.
(b) Let x ft be the height of the overall pen and y ft the width. The general situation is
(c) The total area is A = xy.
(d) The variables are related by 5x + 2y = 750.
(e) We find that,
A(x) = 375 5x.
Setting A(x) = 0, we will get x = 75.
Note that for x < 75, A(x) > 0 and for x > 75, A(x) < 0. Thus x = 75 is a maximum of A by the first derivative test.
Furthermore, A(75) = 14062.5 ft2, which is larger than my estimate in part (a).