Let's set up the linear programming problem based on the given information.

Objective:

Maximize the number of points $P$.

Variables:

• $h$: number of homework assignments completed
• $c$: number of household chores completed
• $P$: number of points earned

Objective Function:

Since Ruby earns 100 points for each homework assignment and 50 points for each household chore, the objective function is: $P = 100h + 50c$ Objective: Maximize $P$

Constraints:

1. Time Constraint: Ruby has at most 8 hours (480 minutes) to complete all tasks. Each homework assignment takes 1 hour (60 minutes) and each household chore takes 45 minutes. $60h + 45c \leq 480$ Simplifying this constraint by dividing by 15: $4h + 3c \leq 32$

2. Homework Assignment Constraint: Ruby has no more than 5 homework assignments to complete. $h \leq 5$

3. Non-negativity Constraints: Ruby cannot complete a negative number of tasks. $h \geq 0$ $c \geq 0$

Summary of the Linear Programming Problem:

Maximize: $P = 100h + 50c$

Subject to:

1. $4h + 3c \leq 32$
2. $h \leq 5$
3. $h \geq 0$
4. $c \geq 0$

Would you like me to solve this linear programming problem or provide more details on any part of it?

Related Questions:

1. How do you solve linear programming problems using the graphical method?
2. What is the role of the objective function in a linear programming problem?
3. How do constraints affect the feasible region in linear programming?
4. What methods can be used to solve linear programming problems besides the graphical method?
5. How does changing the coefficients in the objective function affect the optimal solution?

Tip: Always check the feasibility of the solution by verifying if it satisfies all the constraints before concluding it as the optimal solution.