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Chapter 10: Problem 6

Do the exponential expressions represent growth or decay? $$ 0.22(0.04)^{t} $$

Short Answer

Expert verified
Answer: Decay

Step by step solution

01

Identify the base of the exponent

In the given exponential expression, $$0.22(0.04)^{t},$$ the base of the exponent is 0.04 which is the number being raised to the power of t.
02

Compare the base with 1

Now we need to compare the base value (0.04) with 1 to determine if the expression represents growth or decay. Since 0.04 is between 0 and 1 (0 < 0.04 < 1), it indicates the given expression represents decay.
03

Conclusion

The exponential expression $$0.22(0.04)^{t}$$ represents decay because the base of the exponent (0.04) is between 0 and 1.

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Key Concepts

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

Exponential Growth
Exponential growth describes a process where the quantity increases rapidly over time. In mathematical terms, an exponential growth can be expressed as \( a \cdot b^{t} \), where \( a \) is the initial value, \( b \) is the base (greater than 1), and \( t \) is the time factor or exponent.

When the base \( b \) is greater than 1, it indicates that the quantity is multiplying by more than 1 each time period, leading to rapid increases. For example:
  • If the base \( b \) is 1.05, it means every time period, the quantity grows by an additional 5%.
  • The growth curve is often upward and becomes steeper as time progresses.
Exponential growth is commonly seen in populations, finance (like compound interest), and certain technologies. By understanding the base of the exponent, we can predict whether the scenario depicts growth or decline.
Base of Exponent
The base of an exponent in an expression like \( a \cdot b^{t} \) is the number that gets raised to the power of \( t \). It plays a crucial role in determining the behavior of an exponential expression.

The base (\( b \)) can change the outcome of the function in the following ways:
  • **Greater Than 1:** If \( b > 1 \), the expression represents exponential growth.
  • **Equal to 1:** If \( b = 1 \), the expression remains constant; it doesn't grow or decay.
  • **Less Than 1:** If \( 0 < b < 1 \), the expression signifies exponential decay.
Understanding the base helps to quickly determine the type of exponential behavior involved, making it easier to interpret real-world applications involving exponential functions, such as predicting future trends or modeling time-dependent changes.
Comparison with 1
When dealing with exponential expressions, a key step is to compare the base of the exponent with 1. This comparison reveals whether the expression indicates growth, decay, or stability.

Here's how the comparison works:
  • If the base is **greater than 1**, the expression represents growth, as it increases over time.
  • If the base is **equal to 1**, the expression is stable, meaning it neither grows nor decays, staying constant over time.
  • If the base is **less than 1** (and greater than 0), the expression represents decay, as it decreases over time.
In our example, the base \( 0.04 \) is compared with 1. Since \( 0.04 \) is less than 1, it confirms that the given expression is exponential decay. This process helps in identifying the trend and evaluating scenarios effectively, enhancing comprehension of exponential functions in various practical situations.

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Most popular questions from this chapter

A radioactive substance has a 62 day half-life. Initially there are \(Q_{0}\) grams of the substance. (a) How much remains after 62 days? 124 days? (b) When will only \(12.5 \%\) of the original amount remain? (c) How much remains after 1 day?

Between 1994 and \(1999,\) the national health expenditures in the United States were rising at an average of 5.3\% per year. The U.S. health expenditures in 1994 were 936.7 billion dollars. (a) Express the national health expenditures, \(P\), in billions of dollars, as a function of the year, \(t,\) with \(t=0\) corresponding to the year \(1994 .\) (b) Use this model to estimate the national health expenditures in the year 1999. Compare this number to the actual 1999 expenditures, which were 1210.7 billion dollars.

Solve the equations in Exercise given that $$ f(t)=2^{t}, \quad g(t)=3^{t}, \quad h(t)=4^{t} $$ \(2+h(t)=\frac{33}{16}\)

Solve the equations in Problem for \(x\). Your solutions will involve \(u\). \(\left(\frac{1}{2}\right)^{x}=\left(\frac{1}{16}\right)^{u}\)

Decide for what values of the constant \(A\) the equation has (a) A solution (b) The solution \(t=0\) (c) A positive solution \(A-2^{-t}=0\)
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