Sightseeing boat and writing equation

[mathgeek7365]

A sightseeing boat is chartered by a social club at the rate of $10 per person. The club guarantees the boat company a minimum of 150 people. The boat company agrees to reduce the rate for all passengers by 5 cents a person for each additional person over the 150 minimum. Write and solve an equation to find the number of passengers that will yield the boat company the maximum income.

Let y= total income
x= # of passengers
p= price per passenger

Our problem is the logic behind the equations/how to get the equations. Her teacher said that:

-As long as there are at least 150 passengers, p=10-.05(x-150). Which can be simplified to p=17.5-0.05n.
•She does not understand the x-150 part. Or why you multiply it by 10-0.05.

-The total revenue collected is x times the price per passenger (p), so y=x(17.5-0.05x), which in it's expanded form is y= -0.05x²2+17.5x.
•She completely understand everything here.

She can pretty much solve the equation, she just needs to know why these equations work. Thanks for anything you can assist her with.

 

[MarkFL]

...-As long as there are at least 150 passengers, p=10-.05(x-150). Which can be simplified to p=17.5-0.05n.
•She does not understand the x-150 part. Or why you multiply it by 10-0.05...

First, we are not multiplying $(x-150)$ by $10-0.05$...that would look like:

\(\displaystyle p=(10-.05)(x-150)\)

What we are doing is taking the base price of \$10 per passenger, and for each passenger in excess of 150, or $x-150$, we are subtracting 5 cents.