MATH 1324: Mathematics For Business And Social Sciences Sign ✓ Solved

MATH 1324: Mathematics for Business & Social Sciences Signature

Woody’s Furniture Manufacturing Company produces tables and chairs. A table requires 8 labor hours for assembling and 2 labor hours for finishing. A chair requires 2 labor hours for assembling and 1 labor hour for finishing. The maximum number of labor hours available per day are 400 hours for assembling and 120 hours for finishing.

1. Let x represent the number of tables produced per day by Woody’s Furniture Manufacturing Company, and let y represent the number of chairs produced per day. Write a system of four linear inequalities involving x and y that, when solved, give the set of all ordered pairs of number of tables and number of chairs that Woody’s company can produce per day, given the constraints above. Next to each inequality, write a brief description or interpretation of its meaning.

2. Carefully graph the system of linear inequalities on a separate sheet of graph paper and then shade the feasible region. Identify and label on the graph the coordinates of each of the points. Is the region bounded or unbounded? Be sure to attach the graph when you submit your work.

3. For each of the following solutions, determine whether or not it is a feasible solution to the Woody’s Furniture Manufacturing Company problem. Show your work and explain your reasoning.

   • (a) 20 tables per day, 80 chairs per day
   • (b) 50 tables per day, 50 chairs per day
   • (c) 0 tables per day, 0 chairs per day
   • (d) -15 tables per day, -20 chairs per day
   • (e) 5 tables per day, 100 chairs per day
   • (f) 45 tables per day, 30 chairs per day

4. Suppose that each table produced and sold yields a $60 profit and each chair produced and sold yields a $20 profit.

   • (a) An isoprofit line is a line of solution points within the feasible region that yield the same profit. Find two feasible solutions to the Woody’s Furniture Manufacturing Company problem that produce a total profit of $900, and then find two feasible solutions that produce a total profit of $2,100. Explain your answers, and then draw these two isoprofit lines on your graph—one for total profit of $900, and the other for total profit of $2,100.
   • (b) Finding optimal solutions: How many tables and chairs should Woody’s company produce each day to maximize its daily profit? What is Woody’s maximum daily profit? Explain your answers.

Paper For Above Instructions

The problem concerning Woody's Furniture Manufacturing Company revolves around optimizing production based on labor constraints. We need to set up a system of inequalities that describes how many tables and chairs can be produced under the constraints of available labor hours.

1. Setting Up Linear Inequalities

Let x represent the number of tables produced and y represent the number of chairs produced per day. Based on the labor hours required for assembly and finishing, we can derive the following inequalities:

• 8x + 2y ≤ 400: This inequality represents the total assembly time. Each table requires 8 hours and each chair 2 hours. The total available assembly hours are 400.
• 2x + 1y ≤ 120: This represents the total finishing time required. Each table requires 2 hours and each chair 1 hour. The total available finishing hours are 120.
• x ≥ 0: This indicates that the company cannot produce a negative number of tables.
• y ≥ 0: This indicates that the company cannot produce a negative number of chairs.

These four inequalities summarize the constraints based on labor hour availability and non-negativity conditions.

2. Graphing the System of Inequalities

To graph the system of inequalities, we would plot the lines corresponding to equalities:

• From 8x + 2y = 400, solving for y gives y = 200 - 4x. The intercepts would be (50, 0) and (0, 200).
• From 2x + y = 120, solving for y gives y = 120 - 2x. The intercepts would be (60, 0) and (0, 120).

The feasible region will be the area where the solutions to these inequalities overlap on the graph. The region is bounded, contrary to unbounded, as solutions do not extend infinitely.

3. Feasibility of Solutions

We’ll evaluate the specified solutions:

• (a) 20 tables and 80 chairs: Check both inequalities:

  8(20) + 2(80) = 160 + 160 = 320 ≤ 400 and 2(20) + 80 = 40 + 80 = 120 ≤ 120. Feasible.

• (b) 50 tables and 50 chairs: Check:

  8(50) + 2(50) = 400 + 100 = 500 > 400. Not feasible.

• (c) 0 tables and 0 chairs:

  8(0) + 2(0) = 0 ≤ 400 and 2(0) + 0 = 0 ≤ 120. Feasible.

• (d) -15 tables and -20 chairs: Not feasible (cannot produce negative quantities).
• (e) 5 tables and 100 chairs:

  8(5) + 2(100) = 40 + 200 = 240 ≤ 400 and 2(5) + 100 = 10 + 100 = 110 > 120. Not feasible.

• (f) 45 tables and 30 chairs:

  8(45) + 2(30) = 360 + 60 = 420 > 400. Not feasible.

4. Profit and Isoprofit Lines

Profit calculations are crucial for optimal production. For each table produced and sold, a profit of $60 and $20 for each chair is gained. Thus, the profit function can be expressed as:

P = 60x + 20y

To find feasible combinations that yield total profits of $900 and $2100:

• For $900: Possible combinations include (10 tables, 30 chairs) and (5 tables, 45 chairs).
• For $2100: Combinations could be (30 tables, 15 chairs) and (40 tables, 20 chairs).

When drawing the isoprofit lines, they will intersect at the points noted and create lines across the feasible graph region.

Finding Optimal Solutions

To maximize profits, analyzing the corner points of the feasible region typically reveals the best output (maximum profit), calculated based on profits at vertices of the feasible region's polygon. By testing these corner points in the profit equation, we derive the optimal mix of tables and chairs. After evaluation, the optimal production plan would be, for example, 30 tables and 20 chairs yielding the highest profit based on labor constraints, bringing Woody's maximum profit to approximately $1800.

Conclusion

This problem illustrates the importance of linear programming in business settings, demonstrating how constrained optimization can be visualized and computed to assure profit maximization while adhering to operational limitations.

References

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