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A company that charters a boat for tours on the Gulf Islands can sell 200 tickets at \$50 each. For every \$10 increase in the ticket price five few were tickets will be sold.

Represent the number of tickets sold as a function of the selling price.

Represents the revenue as a function of the selling price.

What selling price will provide the maximum revenue, what is the maximum revenue?

What range of prices will provide a revenue greater than \$20,000?

Micheala95  May 13, 2017
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A company that charters a boat for tours on the Gulf Islands can sell 200 tickets at \$50 each. For every \$10 increase in the ticket price five fewer were tickets will be sold.

Represent the number of tickets sold as a function of the selling price.

Represents the revenue as a function of the selling price.

What selling price will provide the maximum revenue, what is the maximum revenue?

What range of prices will provide a revenue greater than \$20,000?

Let x be the number of \$10  increases

The tickets sold as a function of the selling price = (200 - 5x)

The revenue (per ticket) as a function of the selling price  =  (50 + 10x)

The revenue function  R(x)  is  (revenue per ticket) * ( tickets sold)  where x is the number of \$10 increases after the basic price of \$50 per ticket

So we have...

R(x)  =  (200 - 5x) ( 50 + 10x)   =  -50x^2 + 1750x + 10000

The  x value that will maximize the revenue is given by   -1750 / ( 2 * -50)  =  17.5 ....i.e., the price per ticket is increased to \$225

And the maximum revenue at this price  = \$25,312.50

The range of prices that provide more than \$20000 of revenue  are  ≈ 8 < x < ≈ 27....i.e., when the ticket prices are about \$130 to  about \$320   per ticket

See the graph of the revenue function here :  https://www.desmos.com/calculator/a2msenbvd9

CPhill  May 14, 2017
edited by CPhill  May 14, 2017

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