Math Problem Statement

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.)

Solution

We are tasked with maximizing the contractor's profit, PP, from fixing plumbing and drywall repair. The two variables in the problem are:

  • xx = the number of hours spent fixing plumbing,
  • yy = the number of hours spent repairing drywall.

Each hour of fixing plumbing earns the contractor $40, and each hour of repairing drywall earns $90. Therefore, the total profit PP is based on the number of hours worked on each task:

P=40x+90yP = 40x + 90y

This is the objective function that we want to maximize. It expresses the contractor’s profit in terms of the hours spent on plumbing and drywall work.

Objective function:

Maximizing P=40x+90yP = 40x + 90y.

Next, we should define the constraints for this problem based on the given information.

Would you like help setting up and solving the constraints? Here are five related questions that expand on this problem:

  1. What constraints can we form based on the total hours worked for both tasks?
  2. How do preparation times translate into inequality constraints?
  3. How do we graph the feasible region for the constraints?
  4. How do we use the corner-point method to find the maximum profit?
  5. What is the role of linear programming in business optimization problems like this?

Tip: In linear programming, always check both the feasibility of your solution and the corner points of the feasible region for optimal results.

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Math Problem Analysis

Mathematical Concepts

Linear Programming
Algebra
Optimization

Formulas

P = 40x + 90y (Objective function for maximizing profit)
5x + y ≤ 21 (Preparation time constraint)
x + y ≤ 17 (Work hours constraint)

Theorems

Linear Programming Theorem
Feasible Region and Corner Point Method

Suitable Grade Level

Grades 10-12

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