Question

It costs a bus company $225 to run a bus on a ski trip, plus $30/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 the ticket price by $5. How many empty seats should the bus run with to maximize profit on this trip?

Answer 1

where x is the number of passengers,
cost to the bus company: C = 225 + 30x
payment: P = 60x + 5(22 -x) = 55x + 110
so where is P-C greatest?

Answer 2

Hint: When is the slope of that zero? What could you do to find instantaneous slope at any given point? (you're looking for what called a "critical point")

Answer 3

I'll let BTaylor finish that out, but that's just something to be thinking when you see these min/max type of problems :-)

Answer 4

If you were to graph f(x) = (55x+110)-(225+30) on the interval 0<x<22, it maxes out at (22, 435). So it is best to have no empty seats, and they get a $435 profit.

Answer 5

can you go through how you got the profit equation one more time?

Answer 6

the Payment equation:
they get 60 times the number of passengers, so 60x
then for every passenger less than 22, they get $5 more. So 5(22-x). Add them together.
As for the profit as P-C, they earn what the passengers pay, but they have to subtract the costs.

Answer 7

oh ok, i just didn't see how you got the 5(22-x). How can you find max. revenue for this without graphing?

Answer 8

IDK. maybe see that, in the simplified equation f(x) = 25x-115, the slope is positive, so the far right value of x will have the highest y-value.

Answer 9

well in my textbook, its a multiple choice question. So the options for how many empty seats they should have to max. profit are: 8, 6, 10, or 2.

Answer 10

I'd put 10, since that is the largest number.

Answer 11

so subsitute 12 in for x in the equation?

Answer 12

no, 10 for x. Because x = # of passengers.

Answer 13

yea but 10 is referring to 10 empty seats, so 22-10=12 passengers on board

Answer 14

oh. then 2 empty seats, and x=2. (Sorry, misread your post)

Answer 15

thanks alot!

Answer 16

The empty seat = 8!

Answer 17

-s² + 16s + 87 = 0
Thus Max profit occurs at:
s = -16/-2 = 8 empty seats