How Many Empty Seats Maximize Profit on a Ski Trip Minibus?

[hallowon]

Homework Statement

It Costs a bus company $225 to run a minibus on a ski trip, plus $30 per passenger. the bus has seating for 22 passengers, and the company charges $60 per fare if the bus is full. For each empty seat, the company has to increase ticket price by $5. How many empty seats should the bus run with to maximize profit from this trip?

Homework Equations

vertex form
factored form
standardform
cost function
revenue function
profit function

The back of the book says 8 empty seats

The Attempt at a Solution

my equations
C(x)=(225+30x)
R(x)=(60-5x)
P(x)=R(x)-C(x)
So far I've tried doing the equations numerus ways, I've tried multiplying the terms together, doesn't work I think it gave me 3 empty seats. Tried using the profit functions above, still doesn't give me 8 empty seats. I'm guessing it is either one of my functions that is incorrect but I am not sure which

 

[symbolipoint]

You formulated the Rate ( R(x) ) incorrectly. View the passenger rate as Dollars Per Passenger. The company charges 60 dollars per fare if bus is full, or bus takes 22 passengers. How much money is that? 60 dollars per fare multiplied by 22 fares. Now, what happens for each decrement of 1 passenger? This is where you became stuck(?).
Why did you subtract instead of add? Best I could tell, you want R(x)=60+5x, because "increase ticket price by $5 for each empty seat". x=[count of passengers]

 

[Architeuthis Dux]

one.



To maximize profit, we need to find the number of empty seats that will result in the highest value for the profit function. In this case, the profit function is given by P(x) = R(x) - C(x), where R(x) represents the revenue function and C(x) represents the cost function.

We can rewrite the cost function as C(x) = 225 + 30x, where x represents the number of passengers. Similarly, the revenue function can be written as R(x) = 60x - 5x^2. This is because for each empty seat, the company has to increase the ticket price by $5, resulting in a decrease of $5x in revenue.

To find the maximum value of the profit function, we can use the vertex form, which is given by P(x) = a(x-h)^2 + k, where (h,k) represents the coordinates of the vertex. In this case, h represents the number of empty seats and k represents the maximum profit.

To find the value of h, we can use the factored form of the profit function, which is given by P(x) = -5(x-8)(x-14). This means that the vertex occurs at x = (8+14)/2 = 11. Therefore, the maximum profit is achieved when there are 11 passengers on the bus, leaving 22-11 = 11 empty seats.

However, the question asks for the number of empty seats, not the number of passengers. So we need to subtract the number of passengers from the total number of seats, which is 22. Therefore, the bus should run with 22-11 = 11 empty seats to maximize profit. This is different from the answer given in the book, which could be due to a mistake in the calculations or a different approach used.