Determine whether the following equation represents an exponential growth or exponential decay.
[tex]y=0.81 \cdot(1.65)^x[/tex]
A. exponential growth
B. exponential decay
Answer (1)
Identify the base of the exponential function: b = 1.65 .
Check if the base is greater than 1: 1"> 1.65 > 1 .
Conclude that the equation represents exponential growth because the base is greater than 1.
The equation represents exponential growth: $\boxed{exponential growth}
Explanation
Understanding the Problem We are given the equation y = 0.81" , ( 1.65 ) x and asked to determine whether it represents exponential growth or decay.
Identifying Exponential Growth and Decay An exponential function is of the form y = a " , ( b ) x , where a is the initial value and b is the growth/decay factor. If 1"> b > 1 , the function represents exponential growth, and if 0 < b < 1 , the function represents exponential decay.
Determining Growth or Decay In our equation, y = 0.81" , ( 1.65 ) x , we can see that a = 0.81 and b = 1.65 . Since 1"> 1.65 > 1 , the equation represents exponential growth.
Examples
Exponential growth and decay are useful for modeling various real-world phenomena. For example, population growth, compound interest, and the spread of diseases can be modeled using exponential growth functions. On the other hand, radioactive decay, depreciation of assets, and the cooling of an object can be modeled using exponential decay functions. Understanding exponential growth and decay helps us make predictions and informed decisions in these areas.
