To determine the growth rate of the butterfly population in the county park based on the given function [tex]\( B(t) = 137 \times (1.085)^t \)[/tex], we need to analyze the exponential growth term, [tex]\( (1.085)^t \)[/tex].
In this function, the term [tex]\( 1.085 \)[/tex] inside the parentheses is known as the growth factor. The growth rate can be derived from this growth factor by following these steps:
1. Identify the growth factor: From the function, we see that the growth factor is [tex]\( 1.085 \)[/tex].
2. Determine the growth rate: The growth rate can be found by subtracting 1 from the growth factor.
[tex]\[
\text{Growth Rate} = 1.085 - 1
\][/tex]
3. Convert the growth rate to a percentage: The numerical value obtained represents the growth rate as a decimal. To express it as a percentage, we multiply by 100.
[tex]\[
\text{Growth Rate (as a percentage)} = (1.085 - 1) \times 100
\][/tex]
[tex]\[
\text{Growth Rate (as a percentage)} = 0.085 \times 100
\][/tex]
[tex]\[
\text{Growth Rate (as a percentage)} = 8.5
\][/tex]
Thus, the growth rate of the butterfly population in the county park is [tex]\( 8.5\% \)[/tex].
