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?
Accepted Solution
A:
let's look at the demand quantity as a function of price. Th problem tells us that q(250)= 500 the rate of change of q is a constant, thus, q is a line whose slope is -500/5=-100 tickets/ dollar. therefore we shall have: q(p)=5000-100(p-250)=3000-100p
the revenue for each price will be: r(p)=p×q(p)=p(30000-100p)
next we get the total cost which is: 100×q, where q is the number of people who will fly. but q=30000-100p hence c(p)=100(30000-100p)
the profit will be: Profit=revenue-cost =r(p)-c(p)=p(30000-100p)-100(30000-100p) this will be written in quadratic form as: (p-100)(30000-p) next we find p that maximizes the profit function (p-100)(30000-p)=-p²+30100p-3000000 when you take the derivative of the above, the maximum point will be at p=200, this will give us the profit of: p(200)=-200²+30100(200)-3000000=1000000