If exactly 192 people sign up for a charter flight, Leisure World Travel Agency

Question

If exactly 192 people sign up for a charter flight, Leisure World Travel Agency charges $296/person. However, if more than 192 people sign up for the flight (assume this is the case), then each fare is reduced by $1 for each additional person. Hint: Let x denote the number of passengers above 192.
a) find the revenue function
b)Determine how many passengers will result in a maximum revenue for the travel agency.
c)What is the maximum revenue?
d)What would be the fare per passenger in this case?
Thanks

Explanation / Answer

We know that revenue made is the amount of people times the price per person. With x denoting the amount of passengers above 192, there will be 192 + x passengers. Since the fare is reduces by $1 for each person over 192, the fare is reduced by $x for x addition passengers. Thus, the fare for these 192 + x passengers is 292 - x dollars per person. Thus: R(x) = (192 + x)(292 - x). We can use derivatives to find the maximum. However, note that R(x) is a quadratic. Since that is the case, the maximum value (in this case as the x^2 coefficient is negative) of R(x) can be found when x is the average of the two zeroes. Then, since R(x) has zeroes at x = -192 and x = 292, the minimum value of R(x) is when x = (292 - 192)/2 = 50. Thus, the maximum revenue is (192 + 50)(292 - 50) = $58,564 and the fare per person is 292 - 50 = $242.

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