PRECALCULUS:GRAPH...-NASTA ED.(FLORIDA)
PRECALCULUS:GRAPH...-NASTA ED.(FLORIDA)
10th Edition
ISBN: 9780136174318
Author: Demana
Publisher: PEARSON
Expert Solution & Answer
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Chapter 7.4, Problem 47E
Solution

a.

To prove: For any two real number values for f , the two lines are parallel.

The statement has prove.

Given Information:

The function is defined as,

  f=5x+8y

Calculation:

Consider the given function,

Solve the function for y .

  f=5x+8y8y=−5x+fy=18−5x+f=−58x+18f

As the slope of the line is constant. So for any two values of f , the line is parallel as the value of f will changed.

Therefore, the given statement has proved.

b.

To determine: The reason to moves the line further away from the origin as the value of f increase.

The y -intercept is f .

Given Information:

The function is defined as,

  y=−58x+18f

Explanation:

Consider the given function,

  y=−58x+18f

First, Draw the given line for f=0 .

  PRECALCULUS:GRAPH...-NASTA ED.(FLORIDA), Chapter 7.4, Problem 47E , additional homework tip 1

Increse the value of f as 3, and and draw the graph.0

  PRECALCULUS:GRAPH...-NASTA ED.(FLORIDA), Chapter 7.4, Problem 47E , additional homework tip 2

It can be observed that as the value of f is increasing then the line is moving furhter away from the origin. As this is intercept of y . As the value of intercept is increase then the distance will increase btween line and origin.

Hence, the value of f increase then line moving further away from origin.

c.

To explain: The geometric explanation for the region of example 6 must contain a minimum and a maximum value for f .

The closest point from the origin where a line intersects the restrictions area gives the minimum value, while the furthest point gives the maximum value of the objective function.

Given Information:

The objective function and constraint are defined as,

  f=5x+8y

And,

  2x+y≤102x+3y≤14x≥0y≤0

Calculation:

Consider the given information,

  f=5x+8y

And,

  2x+y≤102x+3y≤14x≥0y≤0

Draw the constraints.

  PRECALCULUS:GRAPH...-NASTA ED.(FLORIDA), Chapter 7.4, Problem 47E , additional homework tip 3

The line  5x+8y=0  passes through the origin. As  f  grows, the lines move above the line  5x+8y=0. The equations of these lines are:

  y=−58+f8

The lines  y=−58+f8 ? intersect the restrictions area. The closest point from the origin where a line intersects the restrictions area gives the minimum value, while the furthest point gives the maximum value of the objective function.

Therefore, the required explanation given above.

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Chapter 7.4, Problem 46E
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Chapter 7.4, Problem 48E

Chapter 7 Solutions

PRECALCULUS:GRAPH...-NASTA ED.(FLORIDA)

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