Precalculus: Mathematics for Calculus - 6th Edition

6th Edition

ISBN: 9780840068071

Author: Stewart, James, Redlin, Lothar, Watson, Saleem

Publisher: Cengage Learning

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Vertices Of An Ellipse

In the study of the conic section of the geometry, An ellipse is a plane curve surrounding two focal points, or foci where the set of all locus of points in the plane has their sum of the distances to the two foci is constant. On the ellipse curve, it is…

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Question

Chapter 3.1, Problem 71E

To determine

To find: The dimensions of the field of largest area that a farmer can fence.

Expert Solution & Answer

Answer to Problem 71E

The dimensions of the field of largest area that the farmer can fence is 600 ft by 1200 ft .

Explanation of Solution

Given information:

A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river. The farmer doesn’t fence along the river.

Calculation:

As per given, assume that the length of the rectangular field is equal to x ft .

As the length is fenced two times i.e. length of the fence for these two sides is equal to 2x ft and as the total fence is 2400 ft . So, the breadth of the rectangular field is equal to 2400−2x ft .

Use the formula for the area of the rectangular region.

A=L×BA=x×(2400−2x)A=2400x−2x2

So, the area function of the rectangular region is given by g(x)=2400x−x2 . To maximize the area of the rectangular region graph the area function g(x)=2400x−x2 .

To graph a function y=2400x−2x2 , 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=2400x−2x2 after the symbol Y1 . Enter the keystrokes 2400X−2X^2 .

The display will show the equation,

Y1=2400X−2X^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 71E

Figure (1)

As observed from the graph, the area function has maximum value at x=600 .

So, the length of the rectangular region is equal to 600 ft and the breadth of the rectangular region is equal to 2400−2(600)=1200 ft .

Therefore, the dimensions of the field of largest area that the farmer can fence is 600 ft by 1200 ft .

Chapter 3.1, Problem 70E

Chapter 3.1, Problem 72E

Chapter 3 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

Ch. 3.1 - To put the quadratic function f(x)=ax2+bx+c in...Ch. 3.1 - The quadratic function f(x) = a(x - h)2 + k is in...Ch. 3.1 - The graph of f(x) = 3(x - 2)2 - 6 is a parabola...Ch. 3.1 - The graph of f(x) = -3(x - 2)2 - 6 is a parabola...Ch. 3.1 - Prob. 5E Ch. 3.1 - Prob. 6E Ch. 3.1 - Prob. 7E Ch. 3.1 - Prob. 8E Ch. 3.1 - Prob. 9E Ch. 3.1 - Prob. 10E

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