Let's break down the problem from the image you've provided.

Given:

• The carpenter earns:

  • $90 for each table (denoted as $T$),
  • $180 for each rocking chair (denoted as $C$).

• Time and manufacturing costs:

  • It takes 2 hours to make a table and 5 hours to make a rocking chair.
  • He is limited to 40 working hours per week.
  • The manufacturing cost is $15 per table and $45 per rocking chair.
  • He wishes to keep his total manufacturing costs to no more than $315 per week.

The objective is to maximize revenue $R$.

Part a)

We need to set up inequalities for the constraints and define the objective function.

1. Objective Function: The revenue $R$ is given by: $R = 90T + 180C$

2. Time Constraint: The total time spent making tables and chairs cannot exceed 40 hours. Since it takes 2 hours to make a table and 5 hours to make a chair: $2T + 5C \leq 40$

3. Cost Constraint: The total manufacturing cost for tables and chairs must not exceed $315. Since the cost is $15 per table and $45 per chair: $15T + 45C \leq 315$

Now, filling in the blanks:

• $R = 90T + 180C$ (Objective function)
• $2T + 5C \leq 40$ (Time constraint)
• $15T + 45C \leq 315$ (Cost constraint)

Part b)

To solve the system of inequalities graphically:

1. Place $C$ on the x-axis and $T$ on the y-axis.
2. The lines representing the constraints will be:
   • $2T + 5C = 40$ for the time constraint.
   • $15T + 45C = 315$ for the cost constraint (this simplifies to $T + 3C = 21$).

We solve these lines for the intercepts and intersections to get the corner points of the feasible region:

• Time constraint: $2T + 5C = 40$

  • $T = 0$, $C = 8$ (when $C = 8$, solves $5C = 40$)
  • $C = 0$, $T = 20$ (when $T = 20$, solves $2T = 40$)

• Cost constraint: $T + 3C = 21$

  • $T = 0$, $C = 7$
  • $C = 0$, $T = 21$

We now have the corner points of the feasible region, which are: $(0, 7), (8, 0), (0, 21), (20, 0)$.

Part c)

To determine which corner point maximizes the objective function $R = 90T + 180C$, we substitute the corner points into the objective function:

• For $(T = 0, C = 7)$: $R = 90(0) + 180(7) = 1260$
• For $(T = 20, C = 0)$: $R = 90(20) + 180(0) = 1800$
• For $(T = 0, C = 21)$: $R = 90(0) + 180(21) = 3780$
• For $(T = 20, C = 0)$: $R = 90(20) + 180(0) = 1800$

The point that maximizes revenue is $(0, 21)$, with a maximum revenue of $3780.

Part d)

Thus, the maximum revenue is $\boxed{3780}$.

Do you want further details or have any questions about the solution?

Here are 5 related questions to explore:

1. What if the working hours were limited to 30 hours instead of 40?
2. How does the manufacturing cost constraint impact the solution?
3. Can you solve this problem using the simplex method for linear programming?
4. How would the maximum revenue change if the revenue per table and chair were different?
5. What if the carpenter wanted to minimize cost instead of maximizing revenue?

Tip: When solving linear programming problems, graphing the constraints often helps in visualizing feasible regions and understanding corner points.