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

#1**+2 **

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