Instructor Guidance Example Week Three Assignment Two Variab

Instructor Guidance Example Week Three Assignment2 Variable Inequali

Here is an example of a problem very similar to the one in the Week Three Assignment: Catskills Hammock Company can obtain at most 2000 yards of striped canvas for making its full size and chair size hammocks. A full size hammock requires 10 yards of canvas and the chair size requires 5 yards of canvas. Write an inequality that limits the number of striped hammocks of each type which can be made.

(b) First I must define what variables I will be using in my inequality. Let f = the number of full size hammocks. Let c = the number of chair size hammocks. Since each full size hammock requires 10 yards of canvas, I will use 10f, and since each chair hammock requires 5 yards of canvas, I will use 5c.

The total amount of canvas which can be used is limited to 2000 yards because that is all they can get. Together my inequality will look like this: 10f + 5c ≤ 2000.

(d) If we call f the independent variable (on the horizontal axis) and c the dependent variable (on the vertical axis), then we can graph the equation using the intercepts. The f-intercept is found when c = 0: 10f ≤ 2000, so f ≤ 200. The f-intercept is (200, 0). The c-intercept is found when f = 0: 5c ≤ 2000, so c ≤ 400. The c-intercept is (0, 400). Because this is a “less than or equal to” inequality, the line will be solid, sloping downward as it moves from left to right.

The region of the graph which is relevant to this problem is restricted to the first quadrant, so the shaded section is from the line towards the origin and stops at the two axes.

(e) Consider the point (105, 175) on my graph. It is inside the shaded area which means the company could fill an order of 105 full size hammocks and 175 chair hammocks. If they made up this many items, they would use 105(10) + 175(5) = 1925 yards of striped canvas and have 75 yards left over. Consider the point (150, 125) on the graph. It is outside the shaded area, which means the company could not make up enough of both kinds of hammocks to fill this order. They would run out of canvas before all of them got made. Calculating: 150(10) + 125(5) = 2125 yards of canvas required. Thus, they cannot fill the order. Consider the point (75, 250). This point is right on the line, meaning the company could fulfill this order exactly without any canvas left over. Calculation: 75(10) + 250(5) = 2000 yards of canvas.

(f) If someone calls and submits an order for 120 full size hammocks and 180 chair hammocks, would the company be able to fill this order? On the graph, the point (120, 180) is outside of the shaded area, indicating they cannot make enough striped hammocks. Evaluating: 120(10) + 180(5) = 1200 + 900 = 2100 yards. The company is 100 yards of canvas short of filling this order.

Paper For Above instruction

The problem of determining the maximum number of products that can be produced given limited resources is a classic application of linear inequalities in operational planning and resource management. This specific scenario involving Catskills Hammock Company exemplifies how inequalities can model constraints effectively to aid decision-making and optimize production within resource limitations.

Linear inequalities serve as a mathematical tool to describe the constraints faced by manufacturers or service providers. In this case, the resource constraint is the limited amount of striped canvas available—2000 yards. The variables \(f\) and \(c\) represent the number of full-size and chair-size hammocks, respectively. Their coefficients, 10 and 5, indicate the yards of canvas required per hammock. The inequality \(10f + 5c \leq 2000\) encapsulates the resource constraint succinctly. This inequality delineates the feasible region in the coordinate plane where production combinations are possible without exceeding the resource limit.

Graphing this inequality involves plotting its intercepts: when \(c=0\), the maximum number of full-size hammocks is \(f=200\); when \(f=0\), the maximum number of chair hammocks is \(c=400\). The region of feasible solutions is the area below or on the boundary line connecting these intercepts in the first quadrant. This geometric interpretation facilitates visual understanding of production limits and helps in decision-making regarding feasible product mixes.

Analyzing specific points within and outside this feasible region provides insights into the company's capacity to meet various order sizes. For example, ordering 105 full-size and 175 chair hammocks corresponds to a point well within the feasible region, requiring \(1925\) yards—less than the available 2000 yards. Conversely, an order of 120 full-size and 180 chair hammocks exceeds the resource limit by requiring 2100 yards, thus is infeasible. Points exactly on the boundary, like (75, 250), demonstrate the maximum production capacity without leftover resources.

This modeling process exemplifies how linear inequalities support strategic planning in manufacturing, ensuring that resource constraints are respected while maximizing output. Such models can also be extended with additional constraints, such as labor hours or machinery hours, to create comprehensive decision-making tools. Ultimately, understanding and applying these inequalities enable businesses to optimize resources, meet customer demands, and improve operational efficiency.

References

• Blitzer, R. (2020). Algebra and problem solving. Pearson Education.
• Gordon, L. (2019). Applied linear algebra. Springer.
• Larson, R., & Edwards, B. H. (2019). Elementary linear algebra (11th ed.). Cengage Learning.
• Lay, D. C. (2016). Linear algebra and its applications (5th ed.). Pearson.
• Murphy, K., & Ottesen, S. (2021). Quantitative methods for business. Routledge.
• Ott, E. (2019). Quantitative approaches in operations management. Wiley.
• Sullivan, M., & Wicks, E. (2019). Operations management (8th ed.). Pearson.
• Thompson, B. (2018). Applied business mathematics. McGraw-Hill Education.
• Wood, R. (2020). Mathematics for economics and business. Routledge.
• Zell, A., & Machacek, M. (2022). Computational linear algebra. Springer.