If exactly 170 people sign up for a charter airplane flight

If exactly 170 people sign up for a charter airplane flight, the Leisure World Travel Agency charges $390 per person. However, if more than 170 people sign up for the flight, then each fare is reduced by $1.50 for each additional person beyond 170. Thus, if 174 people sign up, the fare for every passenger is reduced by $6 (4 times $1.50) and so each fare would be $384. Let x represent the number of passengers on the airplane. a. Find a function p(x) that expresses the price each person pays as a function of the number of passengers on the airplane. Explain what you are doing and include the units of your function. b. Find a function R(x) that expresses the revenue that Leisure World Travel Agency will receive as a function of the number of passengers on the airplane. Explain what you are doing and include the units of your function. c. How many passengers will produce maximum revenue for Leisure World. Do this problem in two different ways: i. Using algebra techniques from the first part of the course. Explain what you are doing. ii. Using calculus techniques from the second half of the course. Explain what you are doing. How do you know you have a maximum? d. How much will each passenger pay? e. How much revenue will Leisure World cam? f. This part is extra credit, and will only be awarded if it is perfect. You\'ll need to explain what you are doing. Using the concept of elasticity, find the price that produces unit elasticity. This should be the same answer you got in part d, because the unit elastic price produces maximum revenue.

Solution

e. Let the number of persons on the charter airplane be x. Then p(x), the price that each person pays to Leisure World Travel Agency is given by p(x) = 390 – (x-170)*1.50 i.e. p(x) = 390 -1.5x +255 or, p(x) = 645-1.5x . Then, the revenue R(x) of Leisure World Travel Agency is R(x) = x*p(x) = x(645 -1.5x) or R(x) = -1.5x² +645x.

f. The demand elasticity refers to how sensitive the demand for particular goods or services is to changes in other economic variables, such as the price. Here, x, the number of persons on the charter plane represents the additional demand   for air travel facility provided by Leisure World Travel Agency for the reduction in price of travel by $1.50 per person. Since p(x) = 645-1.5x is the price that a person pays for rravelling, the price that produces unit elasticity is 645-1.5x.