an airline company offers a round-trip group flight from New York to London. If x people sign up for the flight, the cost of each ticket is to be 1000-2x dollars. Find the maximum revenue the airline company can receive from the sale of tickets from the flight?
To find the maximum revenue the airline company can receive from the sale of tickets for the flight, we need to determine the number of people, x, that will maximize the revenue.
The cost of each ticket is given by the equation C = 1000 - 2x dollars, where C represents the cost of a ticket and x represents the number of people who sign up for the flight.
The revenue generated from ticket sales is given by the equation R = Cx, where R represents the revenue and x represents the number of people who sign up for the flight.
To find the maximum revenue, we need to find the value of x that maximizes the revenue function R = Cx.
To do this, we can use calculus and take the derivative of the revenue function with respect to x, and set it equal to zero to find the critical points. The derivative of the revenue function is:
dR/dx = d(Cx)/dx = C + x(dC/dx),
where dC/dx is the derivative of the cost function.
Substituting the cost function C = 1000 - 2x into the derivative, we get:
dR/dx = (1000 - 2x) + x(-2) = 1000 - 2x - 2x = 1000 - 4x.
Setting dR/dx equal to zero, we have:
1000 - 4x = 0.
Solving for x, we get:
4x = 1000,
x = 1000/4,
x = 250.
So, the critical point is x = 250.
Now, we need to check whether this critical point is a maximum or minimum. We can do this by taking the second derivative of the revenue function and evaluating it at the critical point x = 250.
The second derivative of the revenue function is:
d²R/dx² = d(1000 - 4x)/dx = -4.
Since the second derivative is negative, this indicates that x = 250 is a maximum point.
Now, we can calculate the revenue at x = 250 by substituting the value back into the revenue function:
R = Cx,
R = (1000 - 2x)x,
R = (1000 - 2(250))(250),
R = (1000 - 500)(250),
R = 500(250),
R = 125000.
Therefore, the maximum revenue the airline company can receive from the sale of tickets for the flight is $125,000.
r = x (1000-2x)
r = 1000 x -2 x^2
2 x^2 -1000 x = -r
x^2 - 500 x = -(1/2)r
x^2 -500 x + 250^2 = -(1/2)r + 250^2
(x-250)^2 = -(1/2)(r-2(250)^2 )
x = 250
r = 2*250^2 = 125,000