If exactly 196 people sign up for a charter flight Leisure W

If exactly 196 people sign up for a charter flight, Leisure World Travel Agency charges $302/person. However, if more than 196.people sign up for the flight (assume this is the case), then each fare is reduced by SI for each additional person. Let x denote the number of passengers above 196. Find the revenue function R(x). R(x).R (x) = Determine how many passengers will result in a maximum revenue for the travel agency. passengers What is the maximum revenue? $ What would be the fare per passenger in this case? dollars per passenger

Solution

The revenue function R(x) = ( 196 + x )( 302 - x ) = - x² + 106x + 59192

Now we need to maximize the function R(x)

So that for R\'(x) = 0 , we get

R\'(x) = - 2x + 106 = 0 implies x = 53.

Since the second derivative R\"(x) = - 2, we see that R(x) is concave down, and deduce that x=53 gives the absolute maximum of R. Therefore the number of passengers should be 196 + 53 =249. The fare will then be 302 - 53 = $249 passenger and the revenue will be R(53) = ( 196 + 53 )(302 - 53) = $180,774