What price will generate the greatest profit for the airline?
May.13, 2010 in
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An airline sells all tickets for a certain route at the same price. If it charges 250 dollars per ticket it sells 5000 tickets. For every 5 dollars the ticket price is reduced, an extra 500 tickets are sold. It costs the airline a hundred dollars to fly a person.
What price will generate the greatest profit for the airline?
May 13th, 2010 at 2:19 pm
Your goal is first to find the demand function, then find the revenue function from that, then finally the profit function.
First look at demand (quantity), as a function of price. The problem tells you first that Q(250) = 5000. It also tells you the rate of change of Q is constant, so that Q is a line with slope -500/5 = -100 tickets per dollar. So
Q(P) = 5000 – 100 (P-250) = 30000 – 100 P.
Now you have the quantity for each price, and that gives you the revenue for each price:
R(P) = P*Q(P) = P(30000 – 100P).
Finally you need the total cost. The cost is 100 * Q, where Q is the number of people who fly. But Q = 30000 – 100 P, so the cost is
C(P) = 100(30000-100P).
Now finally the profit is revenue minus cost:
Profit = R(P) – C(P) = P(30000-100P) – 100(30000-100P) = (P-100)(30000-P).
So all you need to do is find the P that maximizes this quadratic function. You expand it out and either take the derivative or complete the square (depending what course you’re in). You get that the maximum occurs when P=200, and the profit at P=200 is ONE MILLION DOLLARS.