Problem 1 - Basic Engineering Mathematics

Question

Experimental values of two related quantities \(x\) and \(y\) are shown below: \begin{tabular}{|r|rrrrrr|} \hline\(x\) & \(0.41\) & \(0.63\) & \(0.92\) & \(1.36\) & \(2.17\) & \(3.95\) \\ \(y\) & \(0.45\) & \(1.21\) & \(2.89\) & \(7.10\) & \(20.79\) & \(82.46\) \\ \hline \end{tabular} The law relating \(x\) and \(y\) is believed to be \(y=a x^{b}\), where \(a\) and \(b\) are constants. Verify that this law is true and determine the approximate values of \(a\) and \(b\).

Step-by-Step Solution

Answer

Perform log transformation, conduct linear regression, and verify results: approx. values of \( a \) and \( b \) fit the data.

Step 1: Take the Logarithm of Both Sides

Start with the equation \( y = a x^b \). We take the logarithm of both sides to linearize the equation. This gives \( \log(y) = \log(a x^b) \). Applying the log properties, this simplifies to \( \log(y) = \log(a) + b \cdot \log(x) \). This equation is now in the form of a linear equation \( Y = B + mX \), where \( Y = \log(y) \), \( B = \log(a) \), \( m = b \), and \( X = \log(x) \).

Step 2: Transform Given Data Using Logarithms

Calculate the logarithms \( \log(x) \) and \( \log(y) \) for all given values:- For \( x = 0.41 \), \( \log(0.41) = -0.387 \)- For \( y = 0.45 \), \( \log(0.45) = -0.347 \)And so on for all \(x\) and \(y\) values. Use these to find all corresponding \( \log(x) \) and \( \log(y) \) values.

Step 3: Perform Linear Regression

Using the transformed data \( \log(x) \) and \( \log(y) \), perform a linear regression to find the best fit line. This line will have a slope \( b \) and an intercept \( \log(a) \). Calculate these using standard linear regression formulas or a calculator/computer software.

Step 4: Interpret Regression Results

From the regression, we obtain the slope \( b \), and the intercept \( \log(a) \). Find \( a \) by taking the antilog of the intercept, i.e., \( a = 10^{\log(a)} \).

Step 5: Verify the Model

Using calculated values of \( a \) and \( b \), verify that the equation \( y = a x^b \) fits the original data. Substitute each \( x \) value into the model and check that the calculated \( y \) values closely match the original \( y \) values.

Key Concepts

linearization of equations

When faced with nonlinear relationships, linearization of equations proves to be a powerful technique. It allows us to convert a complex equation into a simpler, linear form that is easier to analyze and interpret. For instance, if we have a power-law relationship represented by the equation \( y = a x^b \), directly determining the constants \( a \) and \( b \) can be challenging. However, by taking the logarithm of both sides, we can express this equation in a linear form: \( \log(y) = \log(a) + b \cdot \log(x) \).
This linear form resembles the equation of a straight line, \( Y = B + mX \), which is much simpler to work with. Here,

• \( Y = \log(y) \)
• \( B = \log(a) \)
• \( m = b \)
• \( X = \log(x) \)

By transforming the equation like this, we have effectively linearized it, making it more manageable for further analysis.

logarithmic transformation

The logarithmic transformation is a mathematical technique used to linearize data. This transformation is particularly helpful when dealing with multiplicative relationships or exponential growth patterns. In the context of the given problem, taking the logarithmic transformation of the equation \( y = a x^b \) allows us to linearize the relationship and convert it into \( \log(y) = \log(a) + b \cdot \log(x) \).
Each data point \( (x, y) \) is transformed into \( (\log(x), \log(y)) \) using this technique. Specifically:

• The natural logarithm or the common logarithm (base 10) can be used, often depending on the context or tools available.
• This process helps in highlighting the linear relationship between the transformed variables.
• Once transformed, linear regression techniques can be applied to find the relationship constants.

Logarithmic transformation is almighty for data that spans several orders of magnitude, effectively stabilizing variance and normalizing the dataset.

curve fitting

Curve fitting involves finding a curve that best represents a set of data points. In our exercise, it's crucial to fit a model that describes the relationship between \( x \) and \( y \) accurately. With logarithmic transformations, curve fitting becomes manageable while dealing with power functions.
The steps usually involve:

• Transforming the data into a linear form through log transformations.
• Applying linear regression to ascertain the slope \( b \) and the intercept \( \log(a) \).
• Using these parameters to determine \( a \) and fit the curve \( y = a x^b \) back into the original scale of measurement.

Once the appropriate curve is fitted, the model allows for predictions and interpretations about the relationship between the variables involved.

linear regression analysis

Linear regression analysis is a statistical method used to estimate the relationships among variables. In the context of logarithmic regression, it involves using the linearized equation \( \log(y) = \log(a) + b \cdot \log(x) \). This form allows us to apply linear regression techniques to understand or predict the behavior of datasets.
Through this analysis:

• The slope of the line \( b \) is related to the power of \( x \) in the original power model.
• The intercept \( \log(a) \) can be exponentiated to yield the constant \( a \) in the original equation.

This process allows for effective modeling and understanding of data that initially appears non-linear. Linear regression analysis is thus a cornerstone in data analysis, vital in numerous scientific and engineering applications, providing essential insights into the underlying trends of complex datasets.