Problem 1 - Basic Engineering Mathematics

Question

Experimental values of \(x\) and \(y\), shown below, are believed to be related by the law \(y=a x^{2}+b\). By plotting a suitable graph verify this law and determine approximate values of \(a\) and \(b\). \begin{tabular}{|l|lcccc|} \hline\(x\) & 1 & 2 & 3 & 4 & 5 \\ \(y\) & \(9.8\) & \(15.2\) & \(24.2\) & \(36.5\) & \(53.0\) \\ \hline \end{tabular}

Step-by-Step Solution

Answer

The law \( y = ax^2 + b \) is verified by plotting \( y \) against \( x^2 \). Approximate values: \( a \approx 2.1 \), \( b \approx 9 \).

Step 1: Understand the Equation

The relationship given is in the form \( y = ax^2 + b \). This means \( y \) is a quadratic function of \( x \) with \( a \) representing the coefficient of \( x^2 \) and \( b \) representing the constant term.

Step 2: Prepare Data for Graphing

The given data includes \( x \) values \( [1, 2, 3, 4, 5] \) and \( y \) values \( [9.8, 15.2, 24.2, 36.5, 53.0] \). To plot a graph that reveals a linear relationship, we calculate \( x^2 \) for each \( x \), resulting in \( x^2 \) values \( [1, 4, 9, 16, 25] \).

Step 3: Plot the Graph

Plot the values of \( x^2 \) on the x-axis against \( y \) on the y-axis. The points to plot will be: (1, 9.8), (4, 15.2), (9, 24.2), (16, 36.5), and (25, 53.0). A straight-line fit should demonstrate the relationship \( y = ax^2 + b \).

Step 4: Determine the Line Equation

Using the plotted points, draw the best fit line through the points on the graph. This line should be roughly straight if the law holds. Compute the slope \( a \) and y-intercept \( b \) of this line using linear regression or estimation.

Step 5: Calculate Slope (a) and Intercept (b)

Calculate the slope \( a \) using the change in \( y \) divided by the change in \( x^2 \) (choose two points on the line). Then calculate \( b \), the y-intercept, which is the point where the line crosses the y-axis.

Step 6: Verify the Results

Verify the values of \( a \) and \( b \) by substituting back into the equation \( y = ax^2 + b \) and checking against the original values to ensure they closely match. Adjust if necessary.

Key Concepts

Experimental Data Analysis

Experimental data analysis involves examining and evaluating data obtained from experiments, often to establish a mathematical relationship between variables. In this exercise, we are given a set of paired data for variables \( x \) and \( y \). Our task is to verify if these data points fit a quadratic relationship described by the equation \( y = ax^2 + b \).

The process begins with understanding the type of relationship expected amongst the variables. Here, \( y \) is expressed as a quadratic function of \( x \). The goal of performing this analysis is to find the coefficients \( a \) and \( b \) that best fit the given data. The key steps include modifying the data to facilitate analysis, plotting graphs, and applying mathematical formulas to confirm the hypothesized model.

When handling experimental data, it's crucial to:

• Record data accurately to minimize errors.
• Identify and eliminate outliers that may distort the results.
• Use appropriate statistical or graphical tools to interpret the data.

By conducting a thorough analysis, we can validate theoretical models and understand complex relationships between variables.

Graph Plotting

Graph plotting is an essential technique in data analysis, which allows for visual representation and interpretation of data. In this exercise, the relationship we wish to explore is between \( x^2 \) and \( y \). This is because plotting \( x^2 \) against \( y \) should reveal a linear pattern if the given relationship \( y = ax^2 + b \) holds true.

To start, we compute the value of \( x^2 \) for each \( x \), resulting in the sequence \( [1, 4, 9, 16, 25] \). These values are plotted on the x-axis, and the corresponding \( y \) values \( [9.8, 15.2, 24.2, 36.5, 53.0] \) are plotted on the y-axis.

Once the points \((x^2, y)\) are plotted, they should roughly form a straight line if the quadratic relationship is accurate. The plot visualizes the data and provides insight into the kind of mathematical model that best fits it.

Consider the following when plotting graphs:

• Always label axes with appropriate units and scales.
• Ensure data points are plotted precisely to reflect the actual dataset.
• Look for patterns or trends that indicate a particular type of relationship.

This visual analysis helps in confirming the model's validity or identifying necessary adjustments.

Linear Relationship Verification

Linear relationship verification is the process of confirming if a dataset follows a linear pattern. In the case of this quadratic regression exercise, verifying the linear relationship involves checking if the plot of \( x^2 \) against \( y \) forms a straight line.

After plotting the data, draw a line of best fit through the points. This line should pass as closely as possible to all the data points. If the data adheres to the predicted model \( y = ax^2 + b \), the alignment of these points along a line suggests the presence of the relationship. The straightness of this line supports the quadratic relationship hypothesis.

To verify, calculate the slope \( a \) and the intercept \( b \) by selecting two points on the line and using the formula for slope \( a = \frac{\Delta y}{\Delta x^2} \). The intercept \( b \) is the point at which the line crosses the y-axis.

Key points to remember when verifying a linear relationship:

• Always check for consistency in the slope across data points.
• Use more data points for accuracy in calculations.
• Confirm the model by recomputing \( y = ax^2 + b \) for each \( x \) and compare them against actual \( y \) values.

This meticulous verification assures confidence in the mathematical model used and its applicability to the data.