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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?

User Peter VdL
by
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2 Answers

1 vote

Answer:

Scenario II i.e. when the price of ticket is $
$245 will generate the greatest profit for the airline

Explanation:

Scenario I


5000 tickets are sold at $
250 per ticket

The total money earned by the flight agency is

Number of tickets x price of each tickets


= 5000 * 250\\


= 1250000 dollars

Scenario II

The price of each ticket is reduced by $
5

The price of new ticket is


250 -5


= 245 dollars

The new number of tickets sold after reducing the price is


5000+ 500\\


5500

The total money earned by the flight agency is

Number of tickets x price of each tickets


5500 * 245\\


1347500 dollars

Scenario II i.e. when the price of ticket is $
$245 will generate the greatest profit for the airline

User Marc Trudel
by
7.7k points
4 votes
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

User Boj
by
6.2k points