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 33E
Solution

To calculate: The minimum and minimum value, checked they exists, of the objective function f .

The common region in the graph is the solution of the inequality.

Given Information:

The system of inequality is,

  Objective function:f=7x+4y5x+y≥60x+6y≥60Constraints:4x+6y≥204x≥0,y≥0

Calculation:

Consider the given system of inequality,

  Objective function:f=7x+4y5x+y≥60x+6y≥60Constraints:4x+6y≥204x≥0,y≥0

Use the graphing calculator to draw the inequality.

  PRECALCULUS:GRAPH...-NASTA ED.(FLORIDA), Chapter 7.4, Problem 33E , additional homework tip 1

The polygonal region of feasible solutions has three vertices. To find these, solve each of the following system of boundary equations:

Write the inequality as equation and find the intersecting point.

  5x+y=60x+6y=604x+6y=204

Find the points from the equation 5x+y=60 and take 0 for x .

  50+y=60y=60

The obtained point is 0,60 .

Find the points from the equation x+6y=60 and take 0 for x .

  0+6y=606y=60y=10

The obtained point is 0,10 .

Find the points from the equation 4x+6y=204 and take 0 for x .

  40+6y=2046y=204y=34

The obtained point is 0,34 .

So, mentioned the point on the graph and shaded the common region.

  PRECALCULUS:GRAPH...-NASTA ED.(FLORIDA), Chapter 7.4, Problem 33E , additional homework tip 2

Find the value of the objective function at the all vertices.

Substitute 0 for x and 60 for y in the objective function.

  f=70+460=240

Substitute 6 for x and 30 for y in the objective function.

  f=76+430=42+120=162

Substitute 48 for x and 2 for y in the objective function.

  f=748+42=336+8=344

Substitute 60 for x and 0 for y in the objective function.

  f=760+40=420+0=420

As the feasible region is unbounded and has  x≥0  and  y≥0 , and conclude that there is no maximum value for the objective function.

Therefore, the minimum value is 162.

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Chapter 7.4, Problem 32E
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Chapter 7.4, Problem 34E

Chapter 7 Solutions

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

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