Explanation Given Information: At the price $ 18 per ticket, attendance average 40 , 000 people per game and for every decrease of the $ 3 adds 10 , 000 people to the average attendance and $ 4.50 are spent by every person on the concessions. Formula used: The concept of Max-Min Principle-2 is taken into consideration which is as follows: Suppose f is a function such that f ' ( x ) exists for every x in an interval l and there is exactly one critical value c in l , for which f ' ( c ) = 0 Then, f ( c ) is absolute maximum value over l if f ' ' ( c ) < 0 Or f ( c ) is absolute minimum value over l if f ' ' ( c ) > 0 Proof: Let x be the amount by which price should be decreased and let R ( x ) be the revenue. Since, there are two sources of revenue, revenue from tickets and revenue from concessions. Therefore, R ( x ) = ( Revenue from tickets ) + ( Revenue from concessions ) = [ ( Number of People ) ⋅ ( Ticket price ) ] + [ ( Number of People ) ⋅ ( 4.50 ) ] Since, for every decrease in x − dollars, the increase in ticket price is 10 , 000 x . Therefore, the increase in ticket sales is 10 , 000 x 3 for every decrease in x dollars. Therefore, R ( x ) = ( 40 , 000 + 10 , 000 3 x ) ( 18 − x ) + ( 40 , 000 + 10 , 000 3 x ) ( 4.50 ) = − 10 , 000 x 2 + 35 , 000 x + 90 , 000 The value of x such that R ( x ) is maximum, differentiate both sides in above equation to get: R ' ( x ) = − 20000 3 x + 35 , 000 This derivative exists for all value of x , thus critical values are only those where R ' ( x ) = 0 Therefore, − 20000 3 x + 35 , 000 = 0 x = 5

Calculus and Its Applications Plus MyLab Math with Pearson eText -- Access Card Package (11th Edition) (Bittinger, Ellenbogen & Surgent, The Calculus and Its Applications Series)

11th Edition

ISBN: 9780133795561

Author: Marvin L. Bittinger, David J. Ellenbogen, Scott J. Surgent

Publisher: PEARSON

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Chapter 2.5, Problem 87E

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To prove: The price per ticket of a football game so as to maximize profit by Edwards University and the number of person who will attend at that price where at the price $18 per ticket, attendance average 40,000 people per game and for every decrease of the $3 adds 10,000 people to the average attendance and $4.50 are spent by every person on the concessions.

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Chapter 2.5, Problem 86E

Chapter 2.5, Problem 88E

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Calculus and Its Applications Plus MyLab Math with Pearson eText -- Access Card Package (11th Edition) (Bittinger, Ellenbogen & Surgent, The Calculus and Its Applications Series)

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Solution Summary: The author explains how the price per ticket of a football game maximizes profit by Edwards University and the number of people who attend at that price.

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