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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

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) Draw several diagrams illustrating the situation, some with shallow, wide pens and some with deep, narrow pens. Find the total areas of these configurations. Does it appear that there is a maximum area? If so, estimate it.

(b) Draw a diagram illustrating the general situation. Introduce 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).

(a)

To determine

To estimate: The area of configurations by using several diagrams of the rectangular plot with different length and width.

Explanation

Given:

A farmer with 750 ft of fencing wants to enclose a rectangular area then divide it into four pens with fencing parallel to one side of the rectangle.

To draw: Several diagrams illustrating the situation with different length of pens for this rectangular area.

Case 1: Draw a rectangle with length 250ft and width 50 ft .

In the figure 1, the total length of the fence is ,

250+250+50+50+50+50+50=750 ft .

Therefore, the area of the rectangular plot is,

length×width=25050=12,500ft2

Case 2:  Draw a rectangle with length 125 ft and width 100 ft.

In the figure 2, the total length of the fence is,

125+125+100+100+100+100+100=750 ft

(b)

To determine

To draw: The diagram illustrating the general solution for part (a)

(c)

To determine

To find:  The total area for the rectangular field of part (b).

(d)

To determine

To find: The equation regarding the figure 1 in part (b)

(e)

To determine

To find:  The total area of the rectangular field in one variable by using part (d).

(f)

To determine

To solve: The problem using calculus and to compare the answer with the estimate in part (a).

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