Sightseeing boat and writing equation

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

My problem is the logic behind the equations/how to get the equations. My 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.
•I do 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.
•I completely understand everything here.

I can solve the equation, I just need to know why these equations work. Thanks for anything you can assist me with.

DanaAJames said:

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

My problem is the logic behind the equations/how to get the equations. My 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.
•I do not understand the x-150 part. Or why you multiply it by 10-0.05.
...

Click to expand...

Take it one step at a time [assuming at least 150 persons]:
(1) Assume no reduction: Since it is given that the cost is $10 per person no matter how many persons there are, the cost c per ticket with no reduction is given by
c = 10
(2) Compute the reduction: There will be n-150 persons over 150 and for each of these the reduction will be $0.05 so the total reduction r in the price of a single ticket for x persons will be
r = 0.05 (x-150)
(3) Put them together: The total price per customer p will then be the no reduction cost minus the reduction or
p = c - r = 10 - 0.05 (x-150) = -0.05 x + 17.5

DanaAJames said:

10 - 0.05(x - 150)

why you multiply it by 10 - 0.05

Click to expand...

You did not type what you were thinking, yes?

(x - 150) is not multiplied by 10 - 0.05 because 10 - 0.05 is 9.95

(x - 150) is multiplied by 0.05; the product is then subtracted from 10

You posted the correct result, so I'm thinking that you misspoke. :cool: