This problem involves optimizing the contractor's profit based on the time they spend on plumbing and drywall repair. To solve this using linear programming, we need to first identify the objective function and constraints.

Variables:

• x = number of hours spent fixing plumbing.
• y = number of hours spent repairing drywall.
• P = profit in dollars.

Step 1: Identifying the Objective Function

The contractor makes:

• $40 per hour for plumbing.
• $90 per hour for drywall repair.

Thus, the profit P is the sum of the earnings from both tasks:

$P = 40x + 90y$

This is the objective function, which needs to be maximized.

Step 2: Constraints

The problem includes the following constraints:

1. Time Constraint: The contractor does not want to work more than 17 hours total for both tasks. $x + y \leq 17$

2. Preparation Time Constraint: Each hour of plumbing requires 5 hours of preparation, and each hour of drywall repair requires 1 hour of preparation. The contractor cannot spend more than 21 hours on preparation: $5x + y \leq 21$

3. Non-negativity Constraint: The contractor cannot work negative hours on either task, so: $x \geq 0 \quad \text{and} \quad y \geq 0$

Objective Function:

$\text{Maximize} \ P = 40x + 90y$

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Let me know if you'd like the complete solution, including finding the optimal values for $x$ and $y$, or if you have any questions.

Here are 5 related questions:

1. How do we graphically represent the constraints for this problem?
2. What is the feasible region in a linear programming problem?
3. How do we calculate the vertices of the feasible region?
4. What does it mean when we say the solution lies at a "corner point"?
5. How do preparation times influence the total hours worked?

Tip: Linear programming problems are often solved by evaluating the objective function at the vertices of the feasible region.