Word problem help

Question

A theater owner charges $3 for each ticket. Currently, only 100 people attend the theater daily. The owner believe that for each $0.10 decrease per ticket, 10 more people will attend. If this is true and the capacity of the theater is 200, what should the price of a ticket be if the owner want to receive $375 daily from ticket sales?

Answer 1

You would want to write a relationship between price and attendance which in turn will give you sales.

 

The number of people who attend depends on the price and the less the price the more people will attend.

 

Let y = the number of people and x = the price.

 

y = (-10/0.10) x + b = -100 x + b

 

100 people attend at the current price of $3 per ticket.

 

100 = -100 * 3 + b  so b = $400

 

So, y = -100 x + 400

 

If the owner wants to fill the theater, y = 200.

 

200 = -100 x + 400

 

-200 = -100 x

 

x = $2

 

To make $375, the number of people required is 375/x.

 

375/x = -100 x + 400

 

375 = -100 x² + 400 x

 

100x² - 400x + 375 = 0

 

Using the quadratic formula, x = $1.50, $2.50.

 

Since the lowest price is $2 when the theater is filled, the price should be $2.50 to make a sale of $375.

Answer 2

Let y be the number of people going to theater

Let x be the number of dimes to deduct from price

 

so,

 

y = 100 + 10x

 

3 - .1x will be the price

 

(100 + 10x)(3 - .1x) = $375

 

expand above to get:

 

x² -20x +75 = 0

 

solve:

 

(x - 5)(x - 15)

 

so, either 5 dimes or 15 dimes to deduct to receive $375 dalily.

 

but we only deduct 5 dimes ($.50) because of the capacity of theater is only 200.

 

price is $2.50 per ticket.