# The function that models the revenue in terms of ticket price. ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071 ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 3.1, Problem 77E

(a)

To determine

## To evaluate: The function that models the revenue in terms of ticket price.

Expert Solution

The function that models the revenue in terms of ticket price is R(x)=x(570003000x) .

Given:

### Explanation of Solution

Given:

A stadium that holds 55,000 spectators, with the ticket price at $10 the average attendance at recent game has been 27,000 . A market survey indicates that for every dollar the ticket price is lowered, attendance is increases by 3000 . Calculation: Let the ticket price is x . The amount of ticket price is lowered is 10x . According to definition of revenue, Revenue=ticketprice×attendence (1) Increase in attendance is 3000(10x) . Attendance in the recent game is 27000 . Total attendance, Totalattendance=27000+3000(10x)=(570003000x) (2) From the equation (1) and (2), The total revenue in terms of ticket price, R(x)=priceofticket×totalattendance=x×(570003000x)=x(570003000x) (3) From the equation (3), R(x)=3000x2+57000x (4) The standard form of function, f(x)=ax2+bx+c (5) The maximum or minimum value of the function occurs at, x=b2a (6) If a>0 , then the minimum value is f(b2a) . If a<0 , then the maximum value is f(b2a) . From the equation (4) and (5), a=3000b=57000 Substitute 3000 for a and 57000 for b in the equation (6), x=570002×(3000)=9.50 The value of x is$9.50 .

The function has maximum value a=3000<0 .

### Explanation of Solution

Given:

A stadium that holds 55,000 spectators, with the ticket price at $10 the average attendance at recent game has been 27,000 . A market survey indicates that for every dollar the ticket price is lowered, attendance is increases by 3000 . Calculation: Let the ticket price be x . The amount of ticket price is lowered is 10x . According to definition of revenue, Revenue=ticketprice×attendence (1) Increase in attendance is 3000(10x) . Attendance in the recent game is 27000 . Total attendance, Totalattendance=27000+3000(10x)=(570003000x) (2) From the equation (1) and (2), The total revenue in terms of ticket price, R(x)=priceofticket×totalattendance=x×(570003000x)=x(570003000x) (3) From the equation (3), R(x)=3000x2+57000x (4) For the no revenue generated, R(x)=0 (5) From the equation (4) and (5), 57000x3000x2=0(570003000x)x=0 For the value of x , 570003000x=03000x=57000x=19,x=0 For the maximum ticket price no revenue generated x=19 . The maximum ticket price for which no revenue generated is$19.00 .

Thus, the maximum ticket price for which no revenue is generated is \$19.00 .

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