Many businesses use linear programming to determine how to achieve the maximum profit for a product that they are selling. While the mathematical models they use may be more complicated than those we are using today, the process is very much the same. Assume that you have just been charged with helping to return a struggling furniture business to profitability. The business creates two pieces of furniture: dressers and tables. When graphing, the number of tables should be represented on the y-axis while the number of dressers should be on the x-axis. Begin by taking the following statements and translating them into equations or inequalities for questions 1-3. 1.Dressers take five hours to build, while tables take three hours to build. The total number of hours must be less than or equal to 400. 2.The total profit is determined by dressers selling for $325 in profit, while tables sell for $125 in profit. 3.Dressers cost $100 and tables cost $75 for materials manufacture. Total materials cost must not be more than $9,750.

Respuesta :

In order to help return the struggling furniture business to profitability using linear programming, we need to translate the given statements into equations or inequalities:

1. **Time Constraints:**

- Dressers take 5 hours to build, and tables take 3 hours to build.

- Let \( x \) be the number of dressers and \( y \) be the number of tables.

- The total number of hours must be less than or equal to 400:

\( 5x + 3y \leq 400 \)

2. **Profit Calculation:**

- Dressers sell for $325 in profit, and tables sell for $125 in profit.

- The total profit is given by \( 325x + 125y \).

3. **Cost Constraints:**

- Dressers cost $100 and tables cost $75 for materials manufacture.

- The total materials cost must not be more than $9,750:

\( 100x + 75y \leq 9750 \)

By setting up and solving these equations and inequalities simultaneously, the business can optimize the production of dressers and tables to maximize profit while staying within the constraints of time and material costs. This process will involve graphing these constraints on a graph to identify the feasible region and find the optimal solution for the business to achieve maximum profitability.