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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

An employee signs a contract for a salary \(t\) years in the future given in thousands of dollars by \(45(1.041)^{t}\). (a) What do the numbers 45 and 1.041 represent in terms of the salary? (b) What is the salary in 15 years? 20 years? (c) By trial and error, find to the nearest year how long it takes for the salary to double.

Find values for the constants \(a, b\), and \(T\) so that the quantities described are represented by the function \(Q=a b^{-t / T}\). A commercial property initially worth \(\$ 120,000\) loses \(25 \%\) of its value every two years.

(a) Construct a table of values for the function \(y=\) \(253(2.65)^{x}\) for \(x=-3.5,-1.5,0.5,2.5\) (b) At which \(x\) -values in your table is \(253(2.65)^{x} \geq\) \(58.648 ?\)

Find values for the constants \(a, b\), and \(T\) so that the quantities described are represented by the function \(Q=a b^{-t / T}\). A lake begins with 250 fish of a certain species, and one-half disappear every six years.

The value of an investment grows by \(0.06 \%\) every day. By what percent does it increase in a year?
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