Solve the linear programming problem.
-An airline with two types of airplanes, $\mathrm { P } _ { 1 }$ and $\mathrm { P } _ { 2 }$, 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 $P _ { 1 }$ costs $\$ 10,000$ to operate and can accommodate 20 first class, 50 tourist class, and 110 economy class passengers. Airplane $P _ { 2 }$ 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) $11 \mathrm { P } _ { 1 }$ planes and $7 \mathrm { P } _ { 2 }$ planes
B) $5 \mathrm { P } _ { 1 }$ planes and $17 \mathrm { P } _ { 2 }$ planes
C) $9 \mathrm { P } _ { 1 }$ planes and $13 \mathrm { P } _ { 2 }$ planes
D) $7 \mathrm { P } _ { 1 }$ planes and $11 \mathrm { P } _ { 2 }$ planes

Question

Quizplus · Accepted Answer

The objective function is to minimize the total operating cost, which is $10000x_1 + 8500x_2$, where $x_1$ is the number of $P_1$ planes and $x_2$ is the number of $P_2$ planes. The constraints are:- $20x_1 + 18x_2 \ge 400$ (first class)- $50x_1 + 30x_2 \ge 750$ (tourist class)- $110x_1 + 44x_2 \ge 1500$ (economy class)- $x_1, x_2 \ge 0$ (non-negativity)Solving this linear programming problem using a graphical method or a computer program, we get the optimal solution:$x_1 = 9$ (number of $P_1$ planes)$x_2 = 13$ (number of $P_2$ planes)The minimum operating cost is:$10000(9) + 8500(13) = 178500$

Solve the Linear Programming Problem P1\mathrm { P } _ { 1 }P1​

Solve the Linear Programming Problem $\mathrm { P } _ { 1 }$