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Chapter 4: Problem 3

Data that are not exponential: Show that the following data are not exponential. $$ \begin{array}{|c|c|c|c|} \hline t & h(t) & t & h(t) \\ \hline 0 & 4.9 & 3 & 200.2 \\ \hline 1 & 26.6 & 4 & 352.1 \\ \hline 2 & 91.7 & 5 & 547.4 \\ \hline \end{array} $$

Short Answer

Expert verified
The calculated ratios are not constant; hence, the data is not exponential.

Step by step solution

01

Understand Exponential Functions

An exponential function follows the form \( h(t) = a imes b^t \), where \( a \) is the initial amount and \( b \) is the base or growth factor, a constant ratio between consecutive terms.
02

Calculate Ratios of Consecutive Terms

To determine if the data is exponential, calculate the ratios between consecutive values of \( h(t) \). Compute \( \frac{h(t+1)}{h(t)} \) for each consecutive pair: - \( \frac{h(1)}{h(0)} = \frac{26.6}{4.9} \approx 5.43 \)- \( \frac{h(2)}{h(1)} = \frac{91.7}{26.6} \approx 3.45 \)- \( \frac{h(3)}{h(2)} = \frac{200.2}{91.7} \approx 2.18 \)- \( \frac{h(4)}{h(3)} = \frac{352.1}{200.2} \approx 1.76 \)- \( \frac{h(5)}{h(4)} = \frac{547.4}{352.1} \approx 1.55 \)
03

Analyze the Ratios

If the data were exponential, the ratios should be constant. However, the calculated ratios (5.43, 3.45, 2.18, 1.76, 1.55) are not constant, indicating that the data does not follow an exponential form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Data Analysis
Data analysis is a way to interpret and understand information. Imagine you have a list of numbers, just like how we have different values of \( h(t) \) at various times \( t \). By examining these numbers, we can find patterns or trends. This helps in predicting future outcomes or understanding past behavior.

For data that follows an exponential trend, the pattern says that as \( t \) changes, \( h(t) \) should change by a constant factor. In our example, analyzing data helps us discover that the pattern isn't exponential because the growth is not consistent.

By comparing how much \( h(t) \) changes from one point to the next, we can conclude if there鈥檚 a steady pattern or not. If the changes differ like in our example, it's not exponential but rather suggests a varying trend.
Exploring the Growth Factor
The growth factor is a special number in exponential relations. If you have an exponential function \( h(t) = a \times b^t \), the growth factor is \( b \). It tells us how much \( h(t) \) grows between each period \( t \).

For example, if \( b = 2 \), it means each step doubles the previous value. This property is key in exponential functions.

But in the data set here, if you look, every step grows by a different amount: 5.43, 3.45, 2.18, 1.76, 1.55. These values should be the same if the data followed an exponential model. They aren't constant, meaning the data doesn't have a fixed growth factor.
Introduction to Mathematical Modeling
Mathematical modeling involves creating a mathematical representation of a real-world situation. It helps us simplify and analyze complex systems using equations and functions.

In our case, we initially considered modeling the data with an exponential function, hoping \( h(t) = a \times b^t \) matches the data given.

By calculating ratios, we tested our model. However, since the model doesn't align with a single growth factor, it tells us that a different type of model might better explain the data. This is a core process in mathematical modeling: propose, test, and refine your model.
Decoding Consecutive Ratios
Consecutive ratios are calculated by dividing subsequent values in a sequence. For data that is thought to be exponential, these ratios should all be identical. They represent the growth factor over time: \( \frac{h(t+1)}{h(t)} \).

By computing these ratios for the data set
  • \( \frac{26.6}{4.9} \approx 5.43 \)
  • \( \frac{91.7}{26.6} \approx 3.45 \)
  • \( \frac{200.2}{91.7} \approx 2.18 \)
  • \( \frac{352.1}{200.2} \approx 1.76 \)
  • \( \frac{547.4}{352.1} \approx 1.55 \)
we found they are not the same. Non-uniform ratios reveal that data doesn't grow by a consistent factor, proving it鈥檚 not exponential. This analysis helps us select the right model for predicting or understanding data behavior better.

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Most popular questions from this chapter

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Radioactive decay: A physicist has found the following linear model for the natural logarithm of the amount of a radioactive substance present over time: $$ \ln A=-0.073 t+2.336 . $$ Here \(t\) is time in minutes and \(A\) is the amount in grams. Find an exponential model for the amount.
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