Problem 19 - Contemporary Precalculus

Question

Determine whether an exponential, power, or logarithmic model (or none or several of these) is appropriate for the data by determining which (if any) of the following sets of points are approximately linear: $$\\{(x, \ln y)\\}, \quad\\{(\ln x, \ln y)\\}, \quad\\{(\ln x, y)\\}$$ where the given data set consists of the points \(\\{(x, y)\\}\) $$\begin{array}{|l|c|c|c|c|c|c|} \hline x & 5 & 10 & 15 & 20 & 25 & 30 \\ \hline y & 17 & 27 & 35 & 40 & 43 & 48 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

Answer: The logarithmic model is the most appropriate for modeling the given data set because, among the three transformed sets, it appears to have the most linear pattern.

Step 1: Transform the given data set using the three models

For each data point \((x, y)\), transform it into the three models: 1. \((x, \ln y)\) for the exponential model 2. \((\ln x, \ln y)\) for the power model 3. \((\ln x, y)\) for the logarithmic model

Step 2: Analyze the transformed sets for linearity

Check if any of the transformed sets are approximately linear. If any set is linear, then the corresponding model is appropriate for the data set.

Step 3: Determine the appropriate model for the data set

If a linear relationship is found in any of the transformed sets, conclude which model (exponential, power, or logarithmic) is appropriate for modeling the given data set. Step 1 results: Given the data set, we will transform it into the three models: 1. Exponential model: \(\\{(x, \ln y)\\} = \\{(5, \ln 17), (10, \ln 27), (15, \ln 35), (20, \ln 40), (25, \ln 43), (30, \ln 48)\\}\) 2. Power model: \(\\{(\ln x, \ln y)\\} = \\{(\ln 5, \ln 17), (\ln 10, \ln 27), (\ln 15, \ln 35), (\ln 20, \ln 40), (\ln 25, \ln 43), (\ln 30, \ln 48)\\}\) 3. Logarithmic model: \(\\{(\ln x, y)\\} = \\{(\ln 5, 17), (\ln 10, 27), (\ln 15, 35), (\ln 20, 40), (\ln 25, 43), (\ln 30, 48)\\}\) Step 2 analysis: Visually inspecting the three transformed sets, none of them appear to be perfectly linear. However, the set \(\\{(\ln x, y)\\}\) seems to resemble a more linear pattern than the other two sets. Step 3 conclusion: Considering the step 2 analysis, none of the sets appears to be perfectly linear. However, since the logarithmic model seems to be the most linear among the three, it could be considered the most appropriate model for the given data set.

Key Concepts

Exponential Models

Exponential models are powerful tools for describing growth or decay processes. In mathematical terms, an exponential model is expressed as \( y = a \times e^{bx} \), where \( a \) and \( b \) are constants, and \( e \) is the base of the natural logarithm.

These models are suitable for data that shows hockey-stick-like growth, meaning they grow slowly at first and then rapidly, or they can describe decay processes like radioactive decay.

• To determine if a dataset fits an exponential model, we translate the points to the form \((x, \ln y)\).
• If the transformed data creates a linear pattern, then the exponential model is appropriate. This means that the variable's rate of increase is proportional to its current value.

In our exercise, the transformation to \((x, \ln y)\) does not show a strong linear pattern. Therefore, an exponential model might not be the best fit for this data.

Power Models

Power models are another type of mathematical modeling, used when the relationship between variables is best described by an equation of the form \( y = a \times x^b \).

This model is particularly useful in scenarios where quantities are related by powers, such as physical laws like gravity or electrical circuits. It is suitable when data shows a consistent proportional relationship on a log-log scale.

• To check for a power model fit, transform the data into \((\ln x, \ln y)\).
• If the transformed data appears linear, a power model is likely suitable for the data.

In the case of our exercise, transforming the data with \((\ln x, \ln y)\) did not result in a clear linear pattern. Hence, the power model doesn't suit the dataset provided.

Logarithmic Models

Logarithmic models are useful when data changes quickly at first and then levels off. These can be described with a logarithmic equation of the form \( y = a + b \ln x \), where \( a \) and \( b \) are constants.

Logarithmic models find applications in various sciences and financial fields, where initial rapid changes taper off. To analyze whether a logarithmic model fits a dataset, we transform the data into \((\ln x, y)\) format.

• If this transformation results in a linearly arranged data set, it indicates that a logarithmic model is a suitable approach.

In our exercise scenario, the transformed set \((\ln x, y)\) was observed to be closer to a linear arrangement compared to other models. Thus, a logarithmic model would be the most appropriate choice to describe the relationship in the given dataset.