Determine whether the function f(t) = 2(2.3)^(2t) represents exponential growth or exponential decay. Identify the percent rate of change.
Understand the Problem
The question is asking to determine if the given function represents exponential growth or decay and to identify the percent rate of change associated with it.
Answer
The function represents exponential growth, and the rate of change is $130\%$.
Answer for screen readers
The function ( f(t) = 2(2.3)^{2t} ) represents exponential growth, and the percent rate of change is ( 130% ).
Steps to Solve
- Identify the base of the exponential function
For the function ( f(t) = 2(2.3)^{2t} ), the base of the exponent is ( 2.3 ).
- Determine growth or decay
If the base ( b > 1 ), the function represents exponential growth. If ( 0 < b < 1 ), it represents exponential decay. Since ( 2.3 > 1 ), it indicates exponential growth.
- Calculate the percent rate of change
The percent rate of change can be found using the formula:
$$ \text{Percent Rate of Change} = (b - 1) \times 100 $$
Substituting the base:
$$ \text{Percent Rate of Change} = (2.3 - 1) \times 100 = 1.3 \times 100 = 130% $$
The function ( f(t) = 2(2.3)^{2t} ) represents exponential growth, and the percent rate of change is ( 130% ).
More Information
Exponential growth occurs when a quantity increases over time by a consistent percentage. A percent rate of change of ( 130% ) indicates that the function grows at a rapid rate.
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