Question

An airline with two types of airplanes, P1 and P2, has contracted with a tour group to provide transportation for a minimum of 400 first class, 750 tourist class, and 1500 economy class passengers. For a certain trip, airplane P1 costs $10,000 to operate and can accommodate 20 first class, 50 tourist class, and 110 economy class passengers. Airplane P2 costs $8500 to operate and can accommodate 18 first class, 30 tourist class, and 44 economy class passengers. How many of each type of airplane should be used in order to minimize the operating cost?

a) 5 P1 planes and 17 P2 planes
b) 11 P1 planes and 7 P2 planes
c) 7 P1 planes and 11 P2 planes
d) 9 P1 planes and 13 P2 planes

Answer 1

In the problem of minimizing airline operating costs while accommodating specific passenger class requirements, the answer is option d: 9 P1 planes and 13 P2 planes provide the needed capacity at the lowest cost.

Step-by-step explanation:

The question involves determining the optimal number of two types of airplanes, P1 and P2, to minimize operating costs while transporting a minimum of 400 first class, 750 tourist class, and 1500 economy class passengers. This is a typical problem of linear programming or optimizing a linear objective function subject to linear inequality constraints.

To solve such a problem, we would typically set up two variables representing the number of P1 and P2 airplanes. Let's denote x and y as the number of P1 and P2 airplanes, respectively. We then translate the passenger requirements into inequalities based on the capacity of each airplane type and set the cost function to be minimized: Cost = 10000x + 8500y.

However, since this is a multiple-choice question, we can check each option to see which meets the passenger requirements at the lowest cost. Option a (5 P1 and 17 P2) doesn't provide enough first or tourist class seats, option b (11 P1 and 7 P2) meets all requirements but not at minimal cost, option c (7 P1 and 11 P2) does not provide enough tourist class seats, and option d (9 P1 and 13 P2) provides sufficient seats in all classes at the least cost. Therefore, option d is the answer: 9 P1 planes and 13 P2 planes should be used.