To express the total profit P in terms of the number n of tickets sold, we need to first find the total revenue R and the total cost C.
When the ticket price is $200, the airline sells 10,000 tickets, so the total revenue is:
R = $200 x 10,000 = $2,000,000
The cost of flying each person is $100, so the total cost is:
C = $100 x n
When the ticket price is reduced by $30, an extra 1,000 tickets are sold. So if the ticket price is $170, the airline sells 11,000 tickets, and the total revenue is:
R = $170 x 11,000 = $1,870,000
The total cost is still:
C = $100 x n
The profit is the difference between revenue and cost, so we have:
P(n) = R - C = $1,870,000 - $100n when the ticket price is $170, and
P(n) = $2,000,000 - $100n when the ticket price is $200.
Therefore, the total profit P in terms of the number n of tickets sold is:
P(n) = { $1,870,000 - $100n if the ticket price is $170,
$2,000,000 - $100n if the ticket price is $200 }.
To express the total profit P in terms of the price p of one ticket, we can use the same approach. When the ticket price is p, the number of tickets sold is:
n = 10,000 + (p - 200)/30 x 1,000
So the total revenue is:
R = p x n = p [10,000 + (p - 200)/30 x 1,000]
Simplifying this expression, we get:
R = 10,000p + (p - 200)/3 x 1,000
The total cost is still:
C = $100 x n
So the profit is:
P(p) = R - C = (10,000p + (p - 200)/3 x 1,000) - $100n
Substituting n = 10,000 + (p - 200)/30 x 1,000, we get:
P(p) = 10,000p + (p - 200)/3 x 1,000 - $100 [10,000 + (p - 200)/30 x 1,000]
Simplifying this expression, we get:
P(p) = (10,000/3 - $3,333.33)p - $266,666.67
Therefore, the total profit P in terms of the price p of one ticket is:
P(p) = { (10,000/3 - $3,333.33)p - $266,666.67 if p ≤ $170,
($2,000,000 - $100n) - $100n if $170 < p ≤ $200,
($2,000,000 - $100n) - ($100/p) x (n - 11,000) if p > $200 }.
Note that when p > $170, we need to adjust the number of tickets sold to account for the extra 1,000 tickets sold for every $30 reduction in price. The formula above reflects this adjustment.