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, 900 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) 14 P1 planes and 7 P2 planes
B) 13 P1 planes and 9 P2 planes
C) 9 P1 planes and 13 P2 planes
D) 5 P1 planes and 22 P2 planes
B) 13 P1 planes and 9 P2 planes
To find the number of each type of airplane that should be used to minimize the operating cost, let's set up an optimization problem.
Let:
x = number of P1 planes
y = number of P2 planes
Objective function:
We want to minimize the operating cost, given by:
Cost = 10000x + 8500y
Constraints:
1. First-class passengers: 20x + 18y ≥ 400
2. Tourist-class passengers: 50x + 30y ≥ 900
3. Economy-class passengers: 110x + 44y ≥ 1500
We also have the non-negativity constraints:
x ≥ 0
y ≥ 0
Now, let's solve this linear programming problem and see which option satisfies all the constraints.
The solution is:
x = 9
y = 13
So, the correct answer is option C) 9 P1 planes and 13 P2 planes.
To solve this problem, we need to set up a system of equations and use linear programming techniques to find the optimal solution.
Let's define:
x = the number of P1 airplanes
y = the number of P2 airplanes
Since each airplane has a specific capacity for each passenger class, we can write the following constraints:
First Class Constraint:
20x + 18y >= 400 (Total number of first class passengers required)
Tourist Class Constraint:
50x + 30y >= 900 (Total number of tourist class passengers required)
Economy Class Constraint:
110x + 44y >= 1500 (Total number of economy class passengers required)
Additionally, we need to ensure that we are not using negative or fractional values for the number of airplanes, so we include the following constraints:
x, y >= 0 (Non-negativity constraint)
Now, let's calculate the operating cost function:
Operating Cost Function:
Cost = 10,000x + 8,500y (Total operating cost for the given number of airplanes)
To minimize the operating cost, we need to find the values of x and y that satisfy all the constraints and minimize the operating cost function.
Now we can solve this linear programming problem using a tool like the simplex method or a graphing approach. However, we can also do it by trial and error method.
By evaluating the given options, we observe that Option D) 5 P1 planes and 22 P2 planes do not satisfy the constraints for the total number of passengers required.
Let's evaluate the remaining options:
A) 14 P1 planes and 7 P2 planes:
First Class passengers = 20 * 14 + 18 * 7 = 400
Tourist Class passengers = 50 * 14 + 30 * 7 = 770
Economy Class passengers = 110 * 14 + 44 * 7 = 1798
B) 13 P1 planes and 9 P2 planes:
First Class passengers = 20 * 13 + 18 * 9 = 394
Tourist Class passengers = 50 * 13 + 30 * 9 = 757
Economy Class passengers = 110 * 13 + 44 * 9 = 1742
C) 9 P1 planes and 13 P2 planes:
First Class passengers = 20 * 9 + 18 * 13 = 402
Tourist Class passengers = 50 * 9 + 30 * 13 = 777
Economy Class passengers = 110 * 9 + 44 * 13 = 1826
By comparing the number of passengers for each option, the only option that satisfies all the constraints is option A) 14 P1 planes and 7 P2 planes.
Therefore, the answer is A) 14 P1 planes and 7 P2 planes, to minimize the operating cost.