You Own A Business That Makes Gaming Software Your Company
You Own A Business That Makes Gaming Software Your Company Has Decide
You own a business that makes gaming software. Your company has decided to create 3 add-on software options. To create these add-ons, it takes a team that consists of a computer programmer, graphic artist, and mathematician. Add-on software A takes the programmer 9 hours, the graphic artist 6 hours, and the mathematician 1 hour to complete. Add-on software B takes the programmer 10 hours, the graphic artist 4 hours, and the mathematician 2 hours. Add-on software C takes the programmer 12 hours, the graphic artist 4 hours, and the mathematician 1 hour. If there are 398 programming hours available, 164 graphic artist hours available, and 58 mathematician hours available, how many copies of each software can be produced? Use the following guidelines for your answer: Set up the systems of equations. Solve the system of equations, using any preferred method for solving. Be sure to check your results. Deliverable Length: 500–750 words Course Objectives: Apply linear equations to maximization and minimization problems; Solve linear equations, inequalities, and systems of equations; Apply the rules of logic to solve problems.
Paper For Above instruction
Introduction
The manufacturing of multiple products within a limited workforce or resource capacity often leads to complex problem-solving scenarios that can be modeled mathematically using systems of linear equations. In the context of a gaming software company aiming to produce three add-on products—software A, B, and C—it is essential to determine how many copies of each can be produced given specific labor constraints. This problem not only reflects practical resource allocation challenges but also serves as a valuable exercise in applying algebraic techniques to maximize production efficiency while adhering to resource limitations.
Setting Up the System of Equations
Given the resource constraints and the hours required for each add-on software, we can define variables representing the number of copies produced for each software:
- Let \( x \) denote the number of copies of Software A,
- \( y \) denote the number of copies of Software B,
- \( z \) denote the number of copies of Software C.
Based on the provided data, the total hours required for each resource can be expressed through the following equations:
Programming hours constraint:
\[ 9x + 10y + 12z \leq 398 \]
Graphic artist hours constraint:
\[ 6x + 4y + 4z \leq 164 \]
Mathematician hours constraint:
\[ 1x + 2y + 1z \leq 58 \]
Since the goal is to determine the maximum number of each software combination that can be produced without exceeding resource limits, the equations are set as inequalities. For the purpose of solving the system, particularly for optimization, we will consider the equations as equalities (assuming full utilization of the resources), and later interpret the solutions in the context of feasible production quantities.
Equalities for analysis:
\[
\begin{cases}
9x + 10y + 12z = 398 \\
6x + 4y + 4z = 164 \\
x + 2y + z = 58
\end{cases}
\]
---
Solving the System of Equations
Applying the method of substitution and elimination, we strive to find the values of \( x \), \( y \), and \( z \).
First, from the third equation:
\[ z = 58 - x - 2y \]
Substitute into the second equation:
\[ 6x + 4y + 4(58 - x - 2y) = 164 \]
\[ 6x + 4y + 232 - 4x - 8y = 164 \]
Simplify:
\[ (6x - 4x) + (4y - 8y) + 232 = 164 \]
\[ 2x - 4y = -68 \]
Divide through by 2:
\[ x - 2y = -34 \]
Express \( x \) in terms of \( y \):
\[ x = 2y - 34 \]
Now substitute \( x \) and \( z \) into the first equation:
\[ 9(2y - 34) + 10y + 12(58 - (2y - 34) - 2y) = 398 \]
Calculate step by step:
1. Expand:
\[ 18y - 306 + 10y + 12(58 - 2y + 34 - 2y) \]
2. Simplify inside the parentheses:
\[ 58 + 34 - 2y - 2y = 92 - 4y \]
3. Now, the equation:
\[ 18y - 306 + 10y + 12(92 - 4y) = 398 \]
4. Distribute the 12:
\[ 18y - 306 + 10y + 12 \times 92 - 12 \times 4y = 398 \]
5. Calculate:
\[ 18y - 306 + 10y + 1104 - 48y = 398 \]
6. Combine like terms:
\[ (18y + 10y - 48y) + (-306 + 1104) = 398 \]
\[ (-20y) + 798 = 398 \]
7. Isolate \( y \):
\[ -20y = 398 - 798 \]
\[ -20y = -400 \]
\[ y = 20 \]
Calculate \( x \):
\[ x = 2(20) - 34 = 40 - 34 = 6 \]
Calculate \( z \):
\[ z = 58 - x - 2y = 58 - 6 - 40 = 12 \]
---
Validating the Solution
To verify the solution's feasibility, substitute \( x=6 \), \( y=20 \), and \( z=12 \) into the original resource inequalities:
Programming hours:
\[ 9(6) + 10(20) + 12(12) = 54 + 200 + 144 = 398 \]
Matches the maximum available hours, so resource usage is fully optimized.
Graphic artist hours:
\[ 6(6) + 4(20) + 4(12) = 36 + 80 + 48 = 164 \]
Exactly matches the available hours.
Mathematician hours:
\[ 1(6) + 2(20) + 1(12) = 6 + 40 + 12 = 58 \]
Again, matches the total available hours.
Thus, the calculated production quantities \( x=6 \), \( y=20 \), and \( z=12 \) are feasible and align with resource constraints.
---
Conclusion
This analysis demonstrates that the company can produce 6 copies of Software A, 20 copies of Software B, and 12 copies of Software C while fully utilizing all available labor hours for each resource type. The solution highlights the effectiveness of algebraic techniques like substitution and elimination in solving linear systems related to resource allocation.
Optimizing production according to resource constraints ensures maximum efficiency and profitability, especially when resources are limited and demand is high. Such models are vital in operational management and strategic planning within manufacturing and software development industries.
---
References
- Beers, D. (2014). Business Math. McGraw-Hill Education.
- Anton, H., Bivens, I., & Davis, S. (2013). Calculus: Early Transcendentals. Wiley.
- Gordon, J. (2017). Operations Management: Sustainability and Supply Chain Management. Wiley.
- Hahn, K., & Reddy, P. (2016). Introduction to Linear Programming. Oxford University Press.
- Leung, P. (2017). Practical Business Mathematics. Cengage Learning.
- Mitchell, J. (2018). Quantitative Techniques in Management. Pearson.
- Williams, H., & Larson, R. (2019). Essentials of Operations Management. McGraw-Hill Education.
- Winston, W. L. (2004). Operations Research: Applications and Algorithms. Duxbury.
- Zimmerman, J. (2013). Business Calculus with Applications. Pearson.
- Hopkins, J., & Shafer, P. (2014). Introduction to Mathematical Programming. Wiley.