A theatre company has 300 season tickets subscribers. The board of directors has decided to raise the price of a season ticket from the current price of $400. A survey of the subscribers has determined that, for every $20 increase in price, 10 subscribes would not renew their season tickets. Create a quadratic model(equation) and use completing the square to determine what price would maximize the revenue from the season tickets?

Question

25-06-2021
Mathematics

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A theatre company has 300 season tickets subscribers. The board of directors has decided to raise the price of a season ticket from the current price of $400. A survey of the subscribers has determined that, for every $20 increase in price, 10 subscribes would not renew their season tickets. Create a quadratic model(equation) and use completing the square to determine what price would maximize the revenue from the season tickets?

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achkalyne100 achkalyne100 · Answer 1 · 2021-06-25T15:47:03+02:00

A theater wants to raise the ticket by $20.

For each $20 increase they lose 10 subscribers.

At what price can they maximize the revenue?

:

Let x = no. $20 raises in price, and no. of 10 subscriber losses

:

Revenue = no. of subscribers * ticket price

r = (300-10x) * (400+20x)

FOIL

r = 120000 + 6000x - 4000x - 200x^2

A quadratic equation

r(x) = -200x^2 + 2000x + 120000

maximum occurs at the axis of symmetry, find that x = -b/(2a)

in this equation a = -200, b=2000

x =

x = +5

:

Max revenue occurs when ticket price raise is $20 * 5 or $100 to $500

You will lose 5*10 = 50 subscribers or 250

so the revenue will be:

500 * 250 = $125,000

:

psm22415 psm22415 · Answer 2 · 2021-11-26T10:55:44+01:00

The required $125,000 price would maximize the revenue from the season tickets.

Given,

Company has total no. of season ticket subscribes = 300

Current price of the ticket = $400

A survey of the subscribers has determined that, for every $20 increase in price, 10 subscribes would not renew their season tickets.

Let x = no. $20 raises in price, and no. of 10 subscriber losses.

Revenue = ( no. of subscriber ) × ( ticket price)

r = ( 300 - 10x ) ( 400 + 20x )

r = 120000 - 6000x - 4000x - 200[tex]x^{2}[/tex]

The quadratic equation is ,

r = -[tex]200x^{2} - 10000x + 120000[/tex]

Maximum occurs at the axis of symmetry,

To find that,

x = [tex]\frac{-b}{2a}[/tex]

In this equation, a = -200, b=2000

[tex]x = \frac{-2000}{2 (-200)} \\\\[/tex]

x = 5

Max revenue occurs when ticket price raise is = $20 × 5 or $100 to $500

Company will lose 5×10 = 50 subscribers or 250

So the revenue will be,

500 × 250 = $125,000

Hence, The required $125,000 price would maximize the revenue from the season tickets.

For more information about the Maxima and Minima click the link given below.

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