Problem 12 - 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. $$\begin{array}{|l|l|l|l|l|l|l|} \hline x & 1 & 3 & 5 & 7 & 9 & 11 \\ \hline y & 3 & 21 & 55 & 105 & 171 & 253 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

Answer: No, the exponential model is not appropriate for the data set, as the calculated ratios of successive y-values are not consistent.

Step 1: Compute the ratios of successive y-values

For each pair of consecutive y-values, we will divide the second value by the first value to find their ratio. This will tell us if the ratios are consistent, which indicates an exponential model.

Step 2: Create a table with the ratios

Create a new table with the computed ratios to analyze their consistency. For example: $$\frac{21}{3} = 7$$ $$\frac{55}{21} \approx 2.62$$ $$\frac{105}{55} \approx 1.91$$ $$\frac{171}{105} \approx 1.63$$ $$\frac{253}{171} \approx 1.48$$ $$\begin{array}{|l|l|l|l|l|l|} \hline x & 3 & 5 & 7 & 9 & 11 \\\ \hline Ratio & 7 & 2.62 & 1.91 & 1.63 & 1.48 \\\ \hline \end{array}$$

Step 3: Analyze the ratios

Now that we have the ratios, we need to determine if they are consistent or not. From the table, we can observe that the ratios are not consistent as they are decreasing in value: 7, 2.62, 1.91, 1.63, and 1.48

Step 4: Conclusion

Since the ratios of successive y-values are not consistent, it indicates that this data does not follow an exponential model. Therefore, an exponential model is not appropriate for this data set.

Key Concepts

Data Analysis

Data analysis is a critical process in determining patterns or models that fit a given set of data. When approaching a set of data, such as the table provided in the exercise, the goal is to find a model that can describe how the values behave. To do this, we often compute various statistics or transformations to help reveal underlying trends or patterns in the data.

In the given example, we have x-y pairs, where x-values are consecutive odd numbers and y-values seemingly follow a specific pattern related to them. Through the analysis, the aim is to determine whether these y-values grow by a fixed ratio, suggesting an exponential relationship. Data analysis enables us to systematically approach this task, using elements such as the ratios of successive values, which quickly give insight into the nature of the data and the possible models that might apply.

Ratios of Successive Values

One of the simplest and most practical ways to explore a data pattern is by computing the ratios of successive values. This approach is particularly useful when evaluating the potential for an exponential model. An exponential growth pattern would result in a consistent ratio between successive values.

For the given data, we calculate the ratio between each successive y-value. Let's break it down with the first few calculations:

• The ratio between 21 and 3 is computed as \( \frac{21}{3} = 7 \)
• The next ratio, between 55 and 21, is determined as \( \frac{55}{21} \approx 2.62 \)
• Continuing this, \( \frac{105}{55} \approx 1.91 \)

These ratios are not consistent, indicating the growth is not exponential. This analysis shows how ratios can provide a straightforward method to quickly assess the type of relationship between data points.

Mathematical Modeling

Mathematical modeling is the art of representing real-world phenomena using mathematical concepts. In the textbook exercise, the challenge is to find if the y-values follow an exponential trend modeled by mathematical expressions of the form \( y = ab^x \), where \( a \) and \( b \) are constants.

By utilizing the computed ratios from the previous section, we can ascertain whether the data aligns with an exponential model or not. Since the ratios of consecutive y-values are decreasing rather than constant, they do not fit this model type.

Mathematical models like exponential functions are very effective in representing growth-related phenomena such as population, financial investments, and radioactive decay, where a constant percentage rate is involved.

Precalculus

Precalculus serves as a foundational subject that prepares students for calculus by covering essential concepts, including function analysis and mathematical modeling. One of its objectives is to expose students to different types of functions like linear, polynomial, exponential, and logarithmic.

In precalculus, students learn to identify which mathematical models appropriately fit given sets of data by using methods such as ratio analysis, among others. This exercise is a perfect example of employing a precalculus technique—calculating and analyzing ratios of successive values—to investigate whether an exponential model is applicable.

Understanding these concepts in precalculus ensures students have the necessary skills to tackle more complex problems in calculus, as they learn to discern patterns, choose suitable functions, and understand the behavior of different types of mathematical models.