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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

 

 

 

cool cool cool

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

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