This question discusses the annual percent increase or decrease in the given model. To solve questions like this, the reader should know about the exponential growth function. Exponential growth is a process that increases the quantity over time. It occurs when the instantaneous rate of change (i.e., derivative) of an amount with respect to time is proportional to quantity itself. Described as a function, a quantity undergoing exponential growth represents an exponential function of time; that is, variable representing time is an exponent (unlike other types of growth, such as quadratic growth).
If proportionality constant is negative, then the quantity decreases over time and is said to undergo exponential decay. A discrete definition region with equal intervals is also called geometric growth or geometric decrease because the function values form a geometric progression.
The formula for the exponential growth function is
\[ f ( x ) = a ( 1 + r ) ^{ x } \]
Where $ f ( x ) $ is the initial growth function.
$ a $ is the initial amount.
$ r $ is the growth rate.
$ x $ is the number of time intervals.
Growth like this is seen in real life activities or phenomena, such as spread of a viral infection, growth of debt due to compound interest, and spread of viral videos.
Expert Answer
Given model
Equation 1 is:
\[ y = 0.35 ( 2.3 ) ^ { x } \]
The exponential growth function is
Equation 2 is
\[ y = A ( 1 + \gamma ) ^ { x } \]
Where $ A $ is the initial amount.
$ \gamma $ is the annual percent.
$ x $ is the number of years.
\[ A = 0.35 \]
\[ 1 + \gamma = 2.3 \]
\[ \Rightarrow \gamma = 2.3 – 1 \]
\[ \Rightarrow \gamma = 1.3 \]
\[ \Rightarrow \gamma = 1.3 \times 100 \% \]
\[ \gamma = 130 \% \]
The annual percent increase is $ 130 \% $.
Numerical Result
The annual percent increase of the model $ y = 0.35 ( 2.3 ) ^ { x } $ is $ 130 \%$.
Example
Find the annual percentage increase or decrease $ y = 0.45 ( 3.3 ) ^ { x } $ models.
Solution
Given model
Equation 1 is
\[ y = 0.45 ( 2.3 ) ^ { x } \]
The exponential growth function is
Equation 2 is
\[ y = A (1 + \gamma ) ^ { x } \]
Where $ A $ is the initial amount.
$ \gamma $ is the annual percent.
$ x $ is the number of years.
By using equation $ 1 $ and $ 2 $.
\[ A = 0.45 \]
\[ 1 + \gamma = 3.3 \]
\[ \Rightarrow \gamma = 3.3 – 1 \]
\[ \Rightarrow \gamma = 2.3 \]
\[\Rightarrow \gamma = 2.3 \times 100 \% \]
\[ \gamma = 230 \% \]
The annual percent increase is $ 230 \% $.