Engr 112 Homework 4 Winter Term This Assignment Is To Be Com

Engr 112homework 4winter Termthis Assignment Is To Be Completed In

This assignment requires analysis and graphical representation of various functions and data sets using spreadsheet software such as Excel or OpenOffice Calc. The tasks include plotting functions, determining extrema, analyzing economic models, fitting data to specific mathematical forms, and estimating work through numerical integration based on stress-strain data.

Paper For Above instruction

Introduction

This comprehensive assignment encompasses multiple core topics in engineering mathematics, including the graphing of functions, optimization, data fitting, and numerical methods. These exercises not only enhance understanding of mathematical modeling but also demonstrate practical applications in engineering contexts such as electronics manufacturing, materials testing, and data analysis.

Problem 1: Graphing a Quadratic Function and Determining Extrema

The first task asks to graph the quadratic function f(x) = 6x² - 18x + 6. Such a graph is a parabola opening upwards because the leading coefficient (6) is positive. Using a spreadsheet software, plot the function over a suitable range of x, for example from -2 to 4, with increment steps of 0.1 or 0.2 for smoothness. Calculating the vertex helps identify the minimum point, which occurs at x = -b/2a in the quadratic formula. Here, a = 6 and b = -18, so the vertex is located at x = -(-18)/(2*6) = 1.5. Plugging this back into the function yields f(1.5) = 6(1.5)² - 18(1.5) + 6 = 6(2.25) - 27 + 6 = 13.5 - 27 + 6 = -7.5. Therefore, the minimum value of the function is -7.5 at x = 1.5.

Problem 2: Analyzing Cost, Revenue, Break-Even, and Profit Maximization

The second problem involves modeling the relationship between product quantity, selling price, and production costs for electronic components. Let n denote the number of thousands of components sold per week. The price per thousand units is p = 0.5n dollars, and the total revenue R(n) = p n = 0.5n n = 0.5n². The cost C(n) per thousand units is given by c = 0.2n² - 20n + 600, making total costs in dollars as C(n) = c * n = (0.2n² - 20n + 600)n = 0.2n³ - 20n² + 600n.

To establish the relationships, compute total selling price and costs for values of n in increments of 5 from 5 to 50. Using a spreadsheet, tabulate the values:

• n = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50
• Total Revenue R(n) = 0.5n²
• Total Cost C(n) = (0.2n² - 20n + 600)n

Plot both R(n) and C(n) versus n on the same graph for visual comparison. The break-even point occurs where revenue equals cost, so solve R(n) = C(n). For this, set 0.5n² = 0.2n³ - 20n² + 600n, then solve for n to find the exact break-even quantity.

Numerically, the break-even point can be approximated either by solving the equation algebraically or by inspecting the plotted curves to identify where they intersect. The maximum profit occurs at the value of n where profit P(n) = R(n) - C(n) is maximized. Use the spreadsheet's data analysis tools or compute P(n) for each n to identify this optimum.

Problem 3: Data Fitting Using a Semilog Plot

The third task involves plotting given data points:

• x: 0, 0.2, 0.4, 0.6, 0.8, 1.0
• y: 1.8, 3.6, 7.3, 14.7, 29.6, 59.6

First, plot these points on a regular (Cartesian) scatter plot and observe the trend. Next, transform the data for a semilog plot by plotting x against log(y) or y against log(x), depending on the suspected functional form. The goal is to determine whether the data fits a model of the form y = bem^x, i.e., exponential decay or growth.

In a semilog plot (logarithmic y-axis), if the points align roughly along a straight line, then the data can be modeled by y = bem^x. To find the parameters b and m, perform a linear regression on the transformed data:

• Take the natural logarithm of y: ln(y) = ln(b) + x * ln(m)

Plot ln(y) versus x; the slope of the line corresponds to ln(m), and the intercept corresponds to ln(b). Solving these yields the parameters for the exponential model.

Problem 4: Fitting Data to a Power Law Function

The fourth problem involves plotting points:

• x: 1.0, 2.0, 5.0, 7.0, 10
• y: 2.2, 15.3, 47.7, 110.4, (value incomplete)

Complete the data if possible; otherwise, analyze the available points. Plot these points on a standard graph and then on a log-log scale to test whether the data conforms to y = bx^m, a power-law relationship.

On a log-log plot, if the points lie on a straight line, the model y = bx^m is appropriate. Similar to the previous problem, transform the data:

• Take logarithms of both x and y: ln(y) = ln(b) + m * ln(x)

Regression of ln(y) on ln(x) provides values for ln(b) and m, from which the power-law model parameters are derived.

Problem 5: Numerical Integration of Stress-Strain Data

The final task involves estimating the work done on a material sample by numerically integrating the stress versus strain data. Displacement (mm) and Force (N) data points are provided, representing a stress-strain curve. The area under this curve corresponds to the energy input, calculated via numerical methods such as the trapezoidal rule or Simpson’s rule.

Using a spreadsheet, input the displacement and force values in columns. Apply the trapezoidal rule by computing the sum:

\[

\text{Work} \approx \sum_{i=1}^{n-1} \frac{(F_i + F_{i+1})}{2} (d_{i+1} - d_{i})

\]

where \(F_i\) is the force at displacement \(d_i\). This summation approximates the integral of force over displacement, giving the work done in Joules (if force is in Newtons and displacement in meters or millimeters converted accordingly).

Conclusion

This multi-faceted assignment integrates graphing, data analysis, mathematical modeling, and numerical integration to develop a comprehensive understanding of engineering data and functions. Through precise plotting and analysis, engineers can infer critical properties such as extrema, relationships, and energy calculations, which are fundamental in design, manufacturing, and materials testing.

References

• Burden, R. L., & Faires, J. D. (2011). Numerical Analysis (9th ed.). Brooks/Cole.
• DeGroot, M. H., & Schervish, M. J. (2012). Probability and Statistics (4th ed.). Pearson.
• Hamming, R. W. (2012). Numerical Methods for Engineers. Dover Publications.
• Lay, D. C. (2012). Linear Algebra and Its Applications. Pearson.
• Molteni, C., & Medici, M. (2018). Engineering Mathematics. Springer.
• Strang, G. (2016). Introduction to Linear Algebra. Wellesley-Cambridge Press.
• Chapra, S. C., & Canale, R. P. (2015). Numerical Methods for Engineers (7th ed.). McGraw-Hill Education.
• Ross, S. M. (2014). Introduction to Probability and Statistics for Engineers and Scientists. Academic Press.
• Weisstein, E. W. (2020). Numerical Integration. Wolfram Research.
• Zwillinger, D. (2014). CRC Standard Mathematical Tables and Formulae. CRC Press.