Precalculus: Mathematics for Calculus - 6th Edition
Precalculus: Mathematics for Calculus - 6th Edition
6th Edition
ISBN: 9780840068071
Author: Stewart, James, Redlin, Lothar, Watson, Saleem
Publisher: Cengage Learning
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Chapter 10, Problem 10P
To determine

To find: The number of sheets should be shipped from the east side store to customer A and customer B and from the west side store to customer A and customer B t minimize the delivery cost.

Expert Solution & Answer
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Answer to Problem 10P

The numbers of sheets are to be shipped from the east side to the customer A is 50 and east side store to the customer B is 25, the numbers of sheets are to be shipped from the west side to the customer A is 0 and west side to the customer B is 45.

Explanation of Solution

Given:

Customer A needs 50 sheets and Customer B needs 70 sheets.

The east side store delivery cost per sheet is $0.50 to Customer A and. $0.60 to the Customer B.

The west side store delivery cost per sheet is $0.40 to Customer A and. $0.55 to the Customer B.

The east side store has 80 sheets and west side store has 45 in stock.

Calculation:

Suppose the number of sheets to be shipped from the east side to the Customer A is x and to the Customer B is y.

Use the given information to make the inequalities and the objective function for the feasible region.

The required information is shown in the table below.

Customer A Customer B Total
East side

x

 y 80
East side 50−x 70−x 45

The objective function is,

Z=0.5x+0.4(50−x)+0.6y+0.55(15−y)=0.9x+1.15y+58.5 ,

The constraint to get the feasible region has shown below.

x+y≤80x+y≥75x≥0,y≥0x≤50,y≤70}

Take the equality in the above equations to find the value of x and y as,

x+y=80 (1)

And,

x+y=75 (2)

Substitute 50 for x in the equation (1),

x+y=8050+y=80y=80−50y=30

Substitute 50 for x in the equation (2),

x+y=7550+y=75y=75−50y=25

Substitute 70 for y in the equation (1),

x+y=80x+70=80x=80−70x=10

Substitute 70 for y in the equation (2),

x+y=75x+70=75x=75−70x=5

The graph of the given constraints is shown below.

Precalculus: Mathematics for Calculus - 6th Edition, Chapter 10, Problem 10P

Figure (1)

The vertices which lies in the feasible region is shown below.

(50,25),(50,30),(10,70),(5,70)

Substitute the 50 for x and 25 for y in the objective function Z=0.9x+1.15y+58.5 ,

Z=0.9x+1.15y+58.5=0.9⋅50+1.15⋅25+58.5=45+28.75+58.5=132.25

Substitute the 50 for x and 30 for y in the objective function Z=0.9x+1.15y+58.5 ,

Z=0.9x+1.15y+58.5=0.9⋅50+1.15⋅30+58.5=45+34.5+58.5=138

Substitute the 10 for x and 70 for y in the objective function Z=0.9x+1.15y+58.5 ,

Z=0.9x+1.15y+58.5=0.9⋅10+1.15⋅70+58.5=9+80.5+58.5=148

Substitute the 0 for x and 16 for y in the objective function Z=0.9x+1.15y+58.5 ,

Z=0.9x+1.15y+58.5=0.9⋅5+1.15⋅70+58.5=4.5+80.5+58.5=143.5

So, all the satisfies these vertices are shown in the table below.

Vertices Z=0.9x+1.15y+58.5
(50,25) 132 (Minimum)
(50,30) 138
(10,70) 1482
(5,70) 143.5

Thus, the minimum delivery cost is 132 among these vertices for the objective function Z=0.9x+1.15y+58.5 with the numbers of sheets are to be shipped from the east side to the customer A is 50 and east side to the customer B is 25, and the numbers of sheets are to be shipped from the west side to the customer A is 0 and west side to the customer B is 45.

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Chapter 10, Problem 9P
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Chapter 10, Problem 11P

Chapter 10 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

Ch. 10.1 - The system of equations {2x+3y=75xy=9 is a system...Ch. 10.1 - Prob. 2ECh. 10.1 - A system of two linear equations in two variables...Ch. 10.1 - The following is a system of two linear equations...Ch. 10.1 - Prob. 5ECh. 10.1 - Prob. 6ECh. 10.1 - Prob. 7ECh. 10.1 - Prob. 8ECh. 10.1 - Prob. 9ECh. 10.1 - Prob. 10E
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