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 $\mathrm { P } _ { 1 }$ 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) $9 \mathrm { P } _ { 1 }$ planes and $13 \mathrm { P } _ { 2 }$ planes
B) $11 \mathrm { P } _ { 1 }$ planes and $7 \mathrm { P } _ { 2 }$ planes
C) $7 \mathrm { P } _ { 1 }$ planes and $11 \mathrm { P } _ { 2 }$ planes
D) $5 \mathrm { P } _ { 1 }$ planes and $17 \mathrm { P } _ { 2 }$ planes

Question

Quizplus · Accepted Answer

To solve this linear programming problem, we set up inequalities based on the requirements for the minimum number of passengers in each class and the capacities of each type of airplane. Let $x$ be the number of $P_1$ planes and $y$ be the number of $P_2$ planes. The objective is to minimize the cost function $C = 10000x + 8500y$.The constraints are:1. For first class: $20x + 18y \geq 400$2. For tourist class: $50x + 30y \geq 750$3. For economy class: $110x + 44y \geq 1500$Solving these inequalities and considering the objective to minimize the cost, the combination of $9 P_1$ planes and $13 P_2$ planes provides the minimum cost while satisfying all passenger requirements.

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

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