An airline offers economy and business class tickets. For the airline to be profitable, it must sell a minimum of 27 business class tickets and a minimum of 30 economy tickets. The company makes a profit of R245 for each economy ticket and R295 for each business class ticket. At most, the plane has a capacity of 155 travellers. The airline would like to maximize their profit.



Let ﻿x﻿﻿ represent the number of economy tickets sold and let ﻿﻿y﻿﻿ be the number of business class tickets sold.

Identify the constraint and objective functions in this problem. Use the table of guidelines below to type your functions. Do not include any spaces when typing your functions.

   Symbol

  How to type this





   <=



 

   < 





   >=



 

   > 

  Constraints: 

Number of economy tickets: 

Number of business class tickets:  

Capacity: 

Objective function: 

Maximize 

Here are the constraint and objective functions based on the problem you provided:

### Constraints:
1. Minimum number of economy tickets: 
   \[ x \geq 30 \]
   Type: `x>=30`

2. Minimum number of business class tickets:
   \[ y \geq 27 \]
   Type: `y>=27`

3. Capacity constraint (total number of travelers):
   \[ x + y \leq 155 \]
   Type: `x+y<=155`

### Objective function:
Maximize profit:  
   \[ P = 245x + 295y \]
   Type: `P=245x+295y`

Let me know if you'd like more details on how to solve this problem or if you have any other questions!

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Here are 5 related questions:
1. How do we graph these constraints on a coordinate plane?
2. How do we solve the system of inequalities to find feasible solutions?
3. What is the significance of the boundary lines in the solution space?
4. How do we determine the optimal solution for maximizing profit?
5. How do corner points affect the optimization of the objective function?

**Tip**: When solving optimization problems, always check the vertices (corner points) of the feasible region, as these often yield the maximum or minimum values for the objective function.

An airline offers economy and business class tickets. For the airline to be profitable, it must sell a minimum of 27 business class tickets and a minimum of 30 economy tickets. The company makes a profit of R245 for each economy ticket and R295 for each business class ticket. At most, the plane has a capacity of 155 travellers. The airline would like to maximize their profit.
Let ﻿x﻿﻿ represent the number of economy tickets sold and let ﻿﻿y﻿﻿ be the number of business class tickets sold.
Identify the constraint and objective functions in this problem.

Learn how to maximize airline profits using linear programming, considering ticket sales and capacity constraints. The problem involves solving inequalities to identify feasible solutions and optimize profits based on the sale of economy and business class tickets. Suitable for students in Grades 10-12.

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