Explanation Given Information: The price quoted is $3000 per spot for The Big Bang Theory and $1000 for American Dad. The budget is of $12000. No more than 50% of the advertisements will occur on the American Dad. Formula used: The steps for Linear Programming problems: 1. To make the corner points are inside the rectangle draw a rectangle large enough. 2. Shade the outside of the rectangle so that a new bounded feasible region can be defined and locate the new corner points. 3. Obtain the optimal solutions by the use of bounded feasible region. 4. If the optimal solution occurs at the one of the corner points, then only LP problem has that corner point as the optimal solution otherwise it has no optimal solution. Calculation: Consider the number of spots of The Big Bang Theory to be x and consider the number of spots of American Dad to be y. So, according to given data, x + y ≥ 30 3000 x + 1000 y = 120000 − x + y ≤ 0 The constraints include that x ≥ 0 , y ≥ 0 because number of spots of The Big Bang Theory and number of spots of American Dad cannot be negative. Maximize exposure E = 1.8 x + 1.5 y subject to the given following constraints x + y ≥ 30 , 3000 x + 1000 y ≤ 120000 , − x + y = 0 and x ≥ 0 , y ≥ 0 . Consider the first constraint equation. x + y = 30 Isolate the variable y , by subtracting x from both sides, x − x + y = 30 − x y = 30 − x To plot the graph of the function, find the ordered pairs of the first constraint equation by substituting different values of x . For x = 0 , substitute 0 for x , y = 30 − x = 30 − x For x = 10 , substitute 10 for x , y = 30 − 10 = 20 For x = 20 , substitute 20 for x , y = 30 − 20 = 10 Construct a table for the ordered pairs of first constraint equation, x 0 10 20 y 30 20 10 Test the first inequality by ( 0 , 0 ) test coordinates. x + y ≥ 30 0 + 0 ≥ ? 30 0 + 0 ≥ ? 30 0 ≥ 30 The inequality is not satisfied. If the test point does not satisfy the inequality then the set of solutions is the line plus the entire region on the other side of the line i.e. away from the test coordinate ( 0 , 0 ) . Consider the second constraint equation, 3 x + y = 120 Isolate the variable y , by subtracting x from both sides, 3 x − 3 x + y = 120 − 3 x y = 120 − 3 x To plot the graph of the function, find the ordered pairs of the second constraint equation by substituting different values of x . For x = 0 , substitute 0 for x , y = 120 − 3 ⋅ 0 = 120 − 0 = 120 For x = 20 , substitute 20 for x , y = 120 − 3 ⋅ 20 = 120 − 60 = 60 For x = 40 , substitute 40 for x , y = 120 − 3 ⋅ 40 = 120 − 120 = 0 Construct a table for the ordered pairs, x 0 20 40 y 120 60 0 Test the second inequality by ( 1 , 1 ) test coordinates. 3 x + y ≤ 120 3 ⋅ 0 + 0 ≤ ? 120 0 + 0 ≤ ? 120 0 < 120 The inequality is satisfied. If the test point satisfies the inequality then the set of solutions is the line plus the entire region on the same side of the line as the test coordinate ( 1 , 1 )

Finite Mathematics and Applied Calculus (MindTap Course List)

7th Edition

ISBN: 9781337274203

Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Chapter 6.2, Problem 39E

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To calculate: The number of spots person should purchase on Big Bang Theory and American Dad to maximize exposure if the Big Bang theory was watched by 1.8 million viewers and American Dad was watched by 1.5 million viewers. There are at least 30 spots that need to be bought on the Tv shows.

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Chapter 6.2, Problem 38E

Chapter 6.2, Problem 40E

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Chapter 6 Solutions

Finite Mathematics and Applied Calculus (MindTap Course List)

Ch. 6.1 - In Exercises 1-26, sketch the region that...Ch. 6.1 - In Exercises 1-26, sketch the region that...Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E Ch. 6.1 - Prob. 7E Ch. 6.1 - In Exercises 1-26, sketch the region that...Ch. 6.1 - Prob. 9E Ch. 6.1 - Prob. 10E

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The number of spots person should purchase on Big Bang Theory and American Dad to maximize exposure if the Big Bang theory was watched by 1.8 million viewers and American Dad was watched by 1.5 million viewers. There are at least 30 spots that need to be bought on the Tv shows.

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To calculate: The number of spots person should purchase on Big Bang Theory and American Dad to maximize exposure if the Big Bang theory was watched by 1.8 million viewers and American Dad was watched by 1.5 million viewers. There are at least 30 spots that need to be bought on the Tv shows.

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