To solve the problem, we need to derive a function that models the exponentially growing population of bats in a cave and determine the monthly rate of change. Here’s the step-by-step process:
1. Initial Population:
The initial population of bats is given as 6600.
2. Annual Growth Rate:
The bats are growing at an annual exponential rate of 6%.
3. Monthly Growth Rate Calculation:
To find the monthly growth rate from the given annual growth rate, we use the formula for converting an annual rate to a monthly rate. This formula is derived from the compound interest formula:
[tex]\[
\text{Monthly Rate} = (1 + \text{Annual Rate})^{\frac{1}{12}} - 1
\][/tex]
4. Rounded Monthly Growth Rate:
The monthly growth rate, correctly rounded to four decimal places, is 0.0049.
5. Monthly Percentage Rate:
To express the monthly growth rate as a percentage, we multiply by 100:
[tex]\[
\text{Monthly Percentage Rate} = \text{Monthly Rate} \times 100
\][/tex]
The monthly percentage rate of change, rounded to the nearest hundredth of a percent, is 0.49%.
6. Exponential Growth Function:
Based on these details, the function that represents the population of bats after [tex]\( t \)[/tex] years can be written as:
[tex]\[
f(t) = 6600 \times (1 + 0.0049)^{12t}
\][/tex]
This equation incorporates the initial population and the monthly growth rate raised to the appropriate power to account for the number of months over [tex]\( t \)[/tex] years.
So, summarizing the answer:
Function:
[tex]\[
f(t) = 6600 \times (1.0049)^{12t}
\][/tex]
Monthly Growth:
The rate of increase per month is 0.49%.
Therefore, the function and the monthly rate of change are:
Function: [tex]\( f(t) = 6600 \times (1.0049)^{12t} \)[/tex]
Growth: [tex]\( 0.49\% \)[/tex] increase per month
