Williams Commuter Air Service realizes a monthly revenue represented by the following function, where R(x) is measured in dollars and the price charged per passenger is x dollars.
R(x) = 4800x - 120x2
(a) Find the marginal revenue R'(x).
R'(x) = ??
(b) Compute the following values.
R'(19) = ??
R'(20) = ??
R'(21) = ??
(c) Based on the results of part (b), what price should the airline charge in order to maximize their revenue?
$ ??
(a) To find the marginal revenue R'(x), we need to take the derivative of the revenue function R(x) with respect to x.
R(x) = 4800x - 120x^2
To find R'(x), we use the power rule for derivatives. The derivative of x^n is nx^(n-1).
So, taking the derivative of R(x), we get:
R'(x) = 4800 - 240x
(b) Now we substitute the given values into the derived function R'(x) to compute R'(19), R'(20), and R'(21).
R'(19) = 4800 - 240(19) = 4800 - 4560 = 240
R'(20) = 4800 - 240(20) = 4800 - 4800 = 0
R'(21) = 4800 - 240(21) = 4800 - 5040 = -240
(c) The price that the airline should charge in order to maximize their revenue can be determined by finding the value of x where the derivative R'(x) equals zero. In other words, we need to find the critical point where the slope of the revenue function is neither positive nor negative.
R'(x) = 4800 - 240x
Setting R'(x) = 0:
4800 - 240x = 0
240x = 4800
x = 4800 / 240
x = 20
So, the airline should charge $20 per passenger in order to maximize their revenue.