BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Solutions

Chapter 3.1, Problem 72E
To determine

To find: The function that models the total area of the four pens and largest possible total area of the four pens.

Expert Solution

Answer to Problem 72E

The function that models the total area of the four pens is equal to g(x)=750x5x22 and largest possible total area of the four pens is equal to 14062.5ft2 .

Explanation of Solution

Given information:

A rancher with 750ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of rectangle(see figure).

  Precalculus: Mathematics for Calculus - 6th Edition, Chapter 3.1, Problem 72E , additional homework tip  1

Calculation:

Assume that the breadth of the rectangular region is equal to xft and as the sides along the breadth are 5 . So, the total fence for the five sides along breadth is equal to 5xft .

The fence left is used to fence 2 sides along the length of rectangular region. The left fence is equal to 7505xft . So, the length of the rectangular region is equal to 7505x2ft .

Use the formula for the area of the rectangular region.

  A=L×BA=(7505x2)×xA=750x5x22

So, the area function of the rectangular region is given by g(x)=750x5x22 . To maximize the area of the rectangular region graph the area function g(x)=750x5x22 .

To graph a function y=750x5x22 , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press Y= key and enter right hand side of the equation y=750x5x22 after the symbol Y1 . Enter the keystrokes (750X5X^2)÷2 .

The display will show the equation,

  Y1=(750X5X^2)/2

Now, press the GRAPH key and TRACE key to produce the graph of given function in standard window as shown in Figure (1).

  Precalculus: Mathematics for Calculus - 6th Edition, Chapter 3.1, Problem 72E , additional homework tip  2

Figure (1)

As observed from the graph, the area function has maximum value equal to 14062.5ft2 at x=75 .

Therefore, the function that models the total area of the four pens is equal to g(x)=750x5x22 and largest possible total area of the four pens is equal to 14062.5ft2 .

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