A contractor does two types of repair work, fixing plumbing issues and drywall repair. Due to other obligations, they never want to work more than 17 hours per week on these two tasks. The contractor has determined for every hour they work on fixing plumbing, they need five hours to prepare, and for every hour they work on repairing walls, they need one hour of preparation time. They cannot spend more than 21 hours each week in preparing for the two tasks.

If the contractor makes $40 an hour fixing plumbing and $90 an hour repairing walls, how many hours should they work per week at each task to maximize their profit?

The variables needed to solve the linear programming problem are as follows.

*x*=number of hours worked fixing plumbing*y*=number of hours worked repairing drywall*P*=profit in dollars

(a)

Find the objective function. Select whether it's maximizing or minimizing in the first answer blank and enter you equation in the second answer blank. (*Hint*: Your equation must be written in the form *P* = , where the variable you are maximizing or minimizing is alone on one side of the equal sign.)

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.

A contractor does two types of repair work, fixing plumbing issues and drywall repair. Due to other obligations, they never want to work more than 17 hours per week on these two tasks. The contractor has determined for every hour they work on fixing plumbing, they need five hours to prepare, and for every hour they work on repairing walls, they need one hour of preparation time. They cannot spend more than 21 hours each week in preparing for the two tasks. If the contractor makes $40 an hour fixing plumbing and $90 an hour repairing walls, how many hours should they work per week at each task to maximize their profit?

Learn how to solve a linear programming problem to maximize profit for a contractor performing plumbing and drywall repairs. The problem involves constraints on working hours, preparation times, and earning rates, using algebraic optimization techniques. Suitable for high school students in Grades 9-12.

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