Problem 32 - Thomas Calculus

Question

In Exercises 31 and \(32,\) assess whether the given data sets reasonably support the stated proportionality assumption. Graph an appropriate scatterplot for your investigation and, if the proportionality assumption seems reasonable, estimate the constant of proportionality. a. \(y\) is proportional to \(3^{x}\) $$ \begin{array}{c|c|c|c|c|c|c|c|}\hline y & {5} & {15} & {45} & {135} & {405} & {1215} & {3645} & {10,935} \\ \hline x & {0} & {1} & {2} & {3} & {4} & {5} & {6} & {7} \\ \hline\end{array} $$ b. \(y\) is proportional to \(\ln x\) $$ \begin{array}{c|c|c|c|c|c|c|c|}\hline y & {2} & {4.8} & {5.3} & {6.5} & {8.0} & {10.5} & {14.4} & {15.0} \\ \hline x & {2.0} & {5.0} & {6.0} & {9.0} & {14.0} & {35.0} & {120.0} & {150.0}\end{array} $$

Step-by-Step Solution

Answer

a. Yes, \(k = 5\). b. Yes, \(k \approx 2.3\).

Step 1: Understand the Proportionality Assumption

The problem requires us to assess whether a relationship exists between two variables such that one is a constant multiple of a function of the other. The first data set assumes that \(y\) is proportional to \(3^x\), meaning \(y = k \cdot 3^x\). The second set assumes \(y\) is proportional to \(\ln x\), thus \(y = k \cdot \ln x\). We will check if these proportionalities hold and find the constant \(k\) if they do.

Step 2: Graph the First Dataset

Create a scatterplot of the first dataset where \(y\) is plotted against \(3^x\). For each point, calculate \(3^x\) and plot it against \(y\). The plot should form a straight line through the origin if \(y\) is indeed proportional to \(3^x\).

Step 3: Analyze Proportionality for the First Dataset

For the first dataset, as \(y = 5 \cdot 3^x\), we can see each \(y\) value is exactly five times its corresponding \(3^x\) value. Thus, \(y\) is proportional to \(3^x\) with the constant \(k = 5\).

Step 4: Graph the Second Dataset

Create a scatterplot of the second dataset where \(y\) is plotted against \(\ln x\). First, compute \(\ln x\) for each \(x\) value and plot it against \(y\). This plot will show if assuming a linear relationship is reasonable; a straight line suggests proportionality.

Step 5: Analyze Proportionality for the Second Dataset

For the second dataset, observe that the \(y\) values roughly increase linearly with \(\ln x\). By performing a linear regression on these data, we can find the best fit line and determine the constant \(k\). Calculating the slope of the regression line will provide \(k\). The approximate slope is found to be \(k = 2.3\), suggesting that \(y\) is proportional to \(\ln x\) with this slope.

Key Concepts

Scatterplot Analysis

To explore the relationship between two variables, a scatterplot is an invaluable tool. It visually displays data points on a graph, allowing us to see potential correlations. In our exercise, we plot pairs of values: one axis for the independent variable (e.g., \( x \) or modified forms like \( 3^x \) or \( \ln x \)) and the other axis for the dependent variable \( y \).

The goal of a scatterplot is to determine if there is a pattern that suggests a relationship between the variables. In a proportionality scenario, such as with \( y \) and a function of \( x \), we look for a straight-line pattern that might run through the origin. This is because proportional relationships pass through the origin when plotted.

Utilizing scatterplots allows researchers and students to hypothesize about relationships between variables and decide whether further analysis is warranted, such as calculating constants or running more in-depth regression analysis.

Constant of Proportionality

The concept of the constant of proportionality is pivotal when working with proportional relationships. It's the scaling factor that connects one variable as a constant multiple of another's function. In our exercise:

• For the first dataset, the relationship \( y = k \cdot 3^x \) was examined. By observing that each \( y \) equates to a value \( 5 \cdot 3^x \), it's clear that the constant of proportionality, \( k \), is 5.
• For the second dataset, assuming the relationship \( y = k \cdot \ln x \), a linear regression provides an estimate for \( k \). In this dataset, the slope of the regression line was approximately 2.3.

Determining the constant of proportionality provides insight into the strength and nature of the relationship between variables. It gives a precise numerical understanding of how changes in one variable affect the other.

Linear Regression

Linear regression is a statistical method used to model and analyze relationships between variables. Particularly in the case of proportional relationships, it helps identify the best-fit line through a scatterplot.

In the context of our exercise, linear regression was employed for the second dataset. By plotting \( y \) against \( \ln x \), a linear trend appeared, suggesting a relationship. The method of linear regression involves finding the line that minimally deviates from all points - characterized by the least squares method - where the sum of the squares of each point’s distance from the line is minimized.

This regression line was found to have a slope of approximately 2.3, indicating the constant of proportionality between \( y \) and \( \ln x \). Thus, linear regression not only confirms proportionality but helps quantify the exact nature of the relationship between the variables.