Calculus Question

Question

With x people on board, a South African airline makes a profit of (984−3x) rands per person for a specific flight. 

A. How many people would the airline prefer to have on board? 

Answer: x=

B. What is the maximum number of passengers that can board such that the airline still profits? 

Answer: x=

Answer 1

Since the airline makes a profit of (984 - 3x) rands per person when there are x people on board; we conclude that they must make a total of (984 - 3x)x rands on any specific flight in which there are x people on board.

That is: P = 984x - 3x^2. This function represents a parabola with a maximum. To find this maximum, we look for the x-coordinate where the derivative of the Profit (P) is zero. The derivative of P: P' = 984 - 6x. This is equal 0 only when x = 164. Therefore, that's when the profit is maximized and that's the answer to the first question. Now we have to find the left-most zero of the function to find when it will stop being profitable. P = (984 - 3x)x is equal to 0 only when x = 0 or when x = 328. That means that the airline will stop making a profit when there are 328 or more people on board.