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 15P

Business Strategy A small software company publishes computer games, educational software, and utility software. Their business strategy is to market a total of 36 new programs each year, at least four of these being games. The number of utility programs published is never more than twice the number of educational programs. On average, the company makes an annual profit of $5000 on each computer game, $8000 on each educational program, and $6000 on each utility program. How many of each type of software should the company publish annually for maximum profit?

Expert Solution & Answer
Check Mark
To determine

To find: The number of computer games, educational program and utility program should the company publish annually for maximum profit.

Answer to Problem 15P

The company publishes the 4 four computer games, 32 educational software and 0 utility programs to get the maximum profit.

Explanation of Solution

Given:

The business strategy of the company to launch the 36 new programs each year and at least four of being games.

The number of utility programs published is not more than twice the number of educational programs.

The company annual profit on each computer games is $5000 , on each educational program is $8000 and on each utility program is $6000 .

Calculation:

Let the number of computer games published in a year is x and number of educational videos published in a year is y.

Then the number of utility program is (36−x−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.

Computer games Educational programs Utility programs
Number of games x y (36−x−y)
Profit $5000 $8000 $6000

The objective function is,

I=5000x+8000y+6000(36−x−y)=5000x+8000y+216000−6000x−6000y=216000−1000x+2000y ,

The constraint to get the feasible region has shown below.

The company makes at least 4 computer games.

x≥4

The number of utility programs published is not more than twice the number of educational programs.

2y≥36−x−yx+3y≥36

And,

36−x−y≥0x+y≤36

And,

x≥0,y≥0

Now, take the equalities of the above inequalities,

x=4 , x=0,y=0

And,

x+3y=36 (1)

And,

x+y=36 (2)

Substitute 4 for x in the equation (1).

4+3y=363y=36−43y=32y=323

The intersection point is (x,y)=(4,323) .

Substitute 4 for x in the equation (2).

4+y=36y=36−4y=32

The intersection point is (x,y)=(4,32) .

Now, get the value of y in terms of x from the equation (3) to find the intersection points of equation (1) and (2),

x+y=36y=36−x (3)

Substitute 36−x for y in equation (1),

x+3y=36x+3(36−x)=36x+108−3x=362x=72

Further solve the above equation,

x=36

Substitute 36 for x in equation (3),

y=36−36=0

The intersection point is (x,y)=(36,0) .

Now, draw the graph of the above equations,

Precalculus: Mathematics for Calculus - 6th Edition, Chapter 10, Problem 15P

Figure (1)

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

(36,0),(4,323),(4,32)

Substitute the 36 for x and 0 for y in the objective function I=216000−1000x+2000y ,

I=216000−1000x+2000y=216000−1000⋅36+2000⋅0=216000−36000=180000

Substitute the 4 for x and 32 for y in the objective function I=216000−1000x+2000y ,

I=216000−1000x+2000y=216000−1000⋅4+2000⋅32=216000−4000+64000=276000

Substitute the 4 for x and 323 for y in the objective function I=216000−1000x+2000y ,

I=216000−1000x+2000y=216000−1000⋅4+2000⋅323=216000−4000+640003=212000+21333.33

Further solve the above equation,

I=233333.33

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

Vertices I=216000−1000x+2000y
(36,0) 180000
(4,323) 233333.33
(4,32) 276000(Maximum)

The maximum interest rate is 276000 with the 4 computer program, 32 educational programs. Then the utility programs is,

36−x−y=36−4−32=0

Thus, a company made 4 computer programs, 32 educational programs and 0 utility programs at maximum interest for the objective function I=216000−1000x+2000y .

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Chapter 10, Problem 14P
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Chapter 10, Problem 16P

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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Admission Fees The admission fee at an amusement...Ch. 10.1 - Gas Station A gas station sells regular gas for...Ch. 10.1 - Fruit Stand A fruit stand sells two varieties of...Ch. 10.1 - Airplane Speed A man flies a small airplane from...Ch. 10.1 - Boat Speed A boat on a river travels downstream...Ch. 10.1 - Nutrition A researcher performs an experiment to...Ch. 10.1 - Coffee Blends A customer in a coffee shop...Ch. 10.1 - Mixture Problem A chemist has two large containers...Ch. 10.1 - Mixture Problem A biologist has two brine...Ch. 10.1 - Investments A woman invests a total of 20,000 in...Ch. 10.1 - Investments A man invests his savings in two...Ch. 10.1 - Prob. 71ECh. 10.1 - Prob. 72ECh. 10.1 - Prob. 73ECh. 10.1 - Area of a Triangle Find the area of the triangle...Ch. 10.1 - Prob. 75ECh. 10.2 - These exercises refer to the following system:...Ch. 10.2 - Prob. 2ECh. 10.2 - Prob. 3ECh. 10.2 - Prob. 4ECh. 10.2 - Prob. 5ECh. 10.2 - Prob. 6ECh. 10.2 - Prob. 7ECh. 10.2 - Prob. 8ECh. 10.2 - Prob. 9ECh. 10.2 - 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Prob. 44RECh. 10 - Prob. 45RECh. 10 - Prob. 46RECh. 10 - Prob. 47RECh. 10 - Prob. 48RECh. 10 - Prob. 49RECh. 10 - Prob. 50RECh. 10 - Prob. 51RECh. 10 - Prob. 52RECh. 10 - Prob. 53RECh. 10 - Prob. 54RECh. 10 - Prob. 55RECh. 10 - Prob. 56RECh. 10 - Prob. 57RECh. 10 - Prob. 58RECh. 10 - Prob. 59RECh. 10 - Prob. 60RECh. 10 - Prob. 61RECh. 10 - Prob. 62RECh. 10 - Prob. 63RECh. 10 - Prob. 64RECh. 10 - Prob. 65RECh. 10 - Prob. 66RECh. 10 - Prob. 67RECh. 10 - Prob. 68RECh. 10 - Prob. 69RECh. 10 - Prob. 70RECh. 10 - Prob. 71RECh. 10 - Prob. 72RECh. 10 - Prob. 73RECh. 10 - Prob. 74RECh. 10 - Prob. 75RECh. 10 - Prob. 76RECh. 10 - Prob. 77RECh. 10 - Prob. 78RECh. 10 - Prob. 79RECh. 10 - Prob. 80RECh. 10 - Prob. 81RECh. 10 - Prob. 82RECh. 10 - Prob. 83RECh. 10 - Prob. 84RECh. 10 - Prob. 85RECh. 10 - Prob. 86RECh. 10 - Prob. 87RECh. 10 - Prob. 88RECh. 10 - Prob. 89RECh. 10 - Prob. 90RECh. 10 - Prob. 91RECh. 10 - Prob. 92RECh. 10 - Prob. 93RECh. 10 - Prob. 94RECh. 10 - Prob. 95RECh. 10 - Prob. 96RECh. 10 - Prob. 97RECh. 10 - Prob. 98RECh. 10 - Prob. 99RECh. 10 - Prob. 100RECh. 10 - Prob. 101RECh. 10 - Prob. 102RECh. 10 - Prob. 103RECh. 10 - Prob. 104RECh. 10 - Prob. 105RECh. 10 - Prob. 106RECh. 10 - Prob. 107RECh. 10 - Prob. 108RECh. 10 - Prob. 109RECh. 10 - Prob. 110RECh. 10 - Prob. 111RECh. 10 - Prob. 112RECh. 10 - Prob. 1TCh. 10 - Prob. 2TCh. 10 - Prob. 3TCh. 10 - Prob. 4TCh. 10 - Prob. 5TCh. 10 - Prob. 6TCh. 10 - Prob. 7TCh. 10 - Anne, Barry, and Cathy enter a coffee shop. Anne...Ch. 10 - Prob. 9TCh. 10 - Prob. 10TCh. 10 - Only one of the following matrices has an inverse....Ch. 10 - Prob. 12TCh. 10 - Prob. 13TCh. 10 - Prob. 14TCh. 10 - Find the maximum and minimum values of the given...Ch. 10 - Prob. 2PCh. 10 - Prob. 3PCh. 10 - Find the maximum and minimum value of the given...Ch. 10 - Making Furniture A furniture manufacturer makes...Ch. 10 - A Housing Development A housing contractor has...Ch. 10 - Hauling Fruit A trucker hauls citrus fruit from...Ch. 10 - Manufacturing Calculators A manufacturer of...Ch. 10 - Shipping Televisions An electronics discount chain...Ch. 10 - Prob. 10PCh. 10 - Packaging Nuts A confectioner sells two types of...Ch. 10 - Feeding Lab Rabbits A biologist wishes to feed...Ch. 10 - Investing in Bonds A woman wishes to invest 12,000...Ch. 10 - Prob. 14PCh. 10 - Business Strategy A small software company...Ch. 10 - Feasible Region All parts of this problem refer to...
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