Many times business will raise the prices of their goods
or services to increase their profit. However, when they raise their
prices, they usually lose some customers. In such situations, the
price at which the "maximum" profit occurs needs to be found.
For Example: An auditorium has seats for 1200 people. For the
past several days, the auditorium has been filled to capacity for
each show. Tickets currently cost $5.00 and the owner wants to
increase the ticket prices. He estimates that for each $0.50
increase in price, 100 fewer people will attend.
Let x = number of $0.50 price increases. Thus 5.00 + 0.50x
represents the single price and 1200 - 100x represents the
number of tickets sold.
Income = (number of tickets sold) (ticket price)
Income = (1200 - 100x)(5.00 +0.50x)
Income = 6000 + 100x - 50x2
Graph, and then determine what ticket price will maximize
the profit?

Since an auditorium has seats for 1200 people, and for the past several days, the auditorium has been filled to capacity for each show, and tickets currently cost $ 5.00 and the owner wants to increase the ticket prices, and he estimates that for each $ 0.50 increase in price, 100 fewer people will attend, to determine which is the price that will maximize profit, the following calculation must be made:

1200 x 5 = 6000
1100 x 5.5 = 6050
1000 x 6 = 6000

Therefore, the price that will maximize profit is $ 5.50.

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