BuyFind

Precalculus: Mathematics for Calcu...

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

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

Answer to Problem 77E

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

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 x .

The amount of ticket price is lowered 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 is computed as follows.

Totalattendance=27000+3000(10x)=(570003000x) (2)

From the equation (1) and (2),

The total revenue in terms of ticket price is,

R(x)=priceofticket×totalattendance=x×(570003000x)=x(570003000x)

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

(b)

To determine

To evaluate: The price that maximizes revenue from ticket sales.

Expert Solution

Answer to Problem 77E

The price that maximizes revenue from ticket sales is $9.50 .

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 .

Thus, the price that maximizes revenue from ticket sales is $9.50 .

(c)

To determine

To evaluate: The maximum ticket price for which no revenue is generated.

Expert Solution

Answer to Problem 77E

The maximum ticket price for which no revenue is generated is $19.00 .

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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