MHB Sightseeing boat and writing equation

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A sightseeing boat is chartered at $10 per person with a guaranteed minimum of 150 passengers, and the price decreases by 5 cents for each additional passenger. The price per passenger is expressed as p = 10 - 0.05(x - 150), which simplifies to p = 17.5 - 0.05x. Total income is calculated using the equation y = x(17.5 - 0.05x), leading to the expanded form y = -0.05x^2 + 17.5x. The discussion focuses on understanding how the equations are derived, particularly the reasoning behind the x - 150 term and the price adjustments. Clarifying that the price adjustment is based on the number of passengers exceeding the minimum helps in grasping the equation's logic.
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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.
 
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mathgeek7365 said:
...-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:

$$p=(10-.05)(x-150)$$

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.
 
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