Problem 11 - Contemporary Precalculus

Question

Compute the ratios of successive entries in the table to determine whether or not an exponential model is appropriate for the data. $$\begin{array}{|l|l|l|l|l|l|l|} \hline x & 0 & 2 & 4 & 6 & 8 & 10 \\ \hline y & 3 & 15.2 & 76.9 & 389.2 & 1975.5 & 9975.8 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

Question: Determine if an exponential model is appropriate for the given data set: $$ \begin{array}{c|c} x & y \\ \hline 1 & 3 \\ 2 & 15.2 \\ 3 & 76.9 \\ 4 & 389.2 \\ 5 & 1975.5 \\ 6 & 9975.8 \end{array} $$ Answer: Yes, the data set is following an exponential model as the ratios of successive y-values are approximately constant (close to a value of about 5).

Step 1: Calculate the ratios of successive y-values

First, we need to compute the ratios for each of the successive y-values. We do this by dividing the successive y-value by the previous y-value. $$ \frac{15.2}{3}, \frac{76.9}{15.2}, \frac{389.2}{76.9}, \frac{1975.5}{389.2}, \frac{9975.8}{1975.5} $$

Step 2: Evaluate the ratios

Now, we will evaluate these ratios and check if they are approximately constant: $$ \frac{15.2}{3} \approx 5.067 \\ \frac{76.9}{15.2} \approx 5.059 \\ \frac{389.2}{76.9} \approx 5.060 \\ \frac{1975.5}{389.2} \approx 5.073 \\ \frac{9975.8}{1975.5} \approx 5.053 $$

Step 3: Determine if the model is appropriate

As we can see, the ratios are approximately constant throughout the data set (these are all close to a value of about 5). This indicates that the data is indeed following an exponential model. So, an exponential model is appropriate for this data set.

Key Concepts

Ratios in Data Analysis

When analyzing data to determine the suitability of using an exponential model, one fundamental approach involves calculating the ratios of successive data points. A ratio measures how much one value changes relative to another and is calculated by dividing two numbers. In the context of exponential growth or decay, the ratio between successive data points should remain approximately constant; this constant ratio is an indicator of exponential relationships. Consider a set of y-values, each representing data at different points. Calculating the ratio for each successive pair of y-values can highlight patterns:

• Start with the first pair, dividing the second value by the first to find the ratio.
• Continue this process for each consecutive pair of data points.
• Assess these calculated ratios for consistency.

By observing whether these ratios stay the same or vary minimally, we can draw conclusions about the underlying patterns our data exhibits. Finding near-constant ratios generally suggests that the data follows an exponential curve, helping in deciding the function type that would represent the data accurately.

Evaluating Consistency of Ratios

After computing the ratios for a dataset, the next step is to evaluate these values to assess whether they are consistent with one another. A crucial part of determining if an exponential model is appropriate is confirming this consistency among the resultant ratios. Consistency here implies that all ratios are close enough, with minimal deviations. Consider the following steps:

• Check if the ratios are close to a specific numerical value throughout the set.
• Observe the uniformity; small variations are often acceptable as data in real-world situations might not perfectly line up due to noise or rounding.
• Use these observations as a key indicator in selecting the appropriate model type.

For instance, in the example solution provided, the ratios varied slightly around 5, suggesting that these variations don't detract from consistency, thus supporting the appropriateness of an exponential model. Evaluating these ratios lends valuable insights into identifying the relationship between variables in data modeling.

Introduction to Mathematical Modeling

Mathematical modeling is a powerful tool used to represent and analyze real-world phenomena using mathematical symbols and expressions. The goal is to capture essential characteristics of a system with a mathematical description that is both useful and accurate. One common type of model is the exponential model, particularly useful for describing systems that grow or decay at a constant rate. The process of mathematical modeling involves:

• Identifying relationships between variables through observational data.
• Utilizing mathematical formulas to represent these relationships.
• Testing model predictions against additional data to verify accuracy.
• Refining the model to enhance its predictive capabilities.

In the context of exponential models, if the calculated ratios of successive data points remain constant, an exponential function can effectively represent the data. This approach allows predictability in understanding how changes in one variable might affect the outcome of another, pivotal for strategic decision-making and forecasting.