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

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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