Absolutely, let's work through this problem step-by-step.
1. Understanding the Problem:
We start with a population of 60 gray seals on an island, and this population increases at an annual growth rate of 15%. We need to find the exponential growth function to describe the population of seals after [tex]\( t \)[/tex] months and determine the monthly growth rate.
2. Conversion of Annual Growth Rate to Monthly Growth Rate:
The given annual growth rate is 15%, which we can express as a decimal: [tex]\( 0.15 \)[/tex].
3. Monthly Growth Rate Calculation:
To find the monthly growth rate, we need to convert the annual growth rate into a monthly rate. The relationship between the annual growth rate (which compounds once a year) and the monthly growth rate (which compounds 12 times a year) can be expressed as follows:
[tex]\[
(1 + r_{\text{annual}}) = (1 + r_{\text{monthly}})^{12}
\][/tex]
Here, [tex]\( r_{\text{annual}} \)[/tex] is 0.15. We solve the above equation for [tex]\( r_{\text{monthly}} \)[/tex]:
[tex]\[
1 + 0.15 = (1 + r_{\text{monthly}})^{12}
\][/tex]
[tex]\[
1.15 = (1 + r_{\text{monthly}})^{12}
\][/tex]
Taking the 12th root of both sides:
[tex]\[
1 + r_{\text{monthly}} = \sqrt[12]{1.15}
\][/tex]
Subtracting 1:
[tex]\[
r_{\text{monthly}} = \sqrt[12]{1.15} - 1
\][/tex]
4. Monthly Growth Rate:
Upon solving the above expression numerically, we find that the monthly growth rate [tex]\( r_{\text{monthly}} \)[/tex] is approximately:
[tex]\[
r_{\text{monthly}} = 0.01171491691985338
\][/tex]
5. Exponential Growth Function:
Now, we can formulate the exponential growth function describing the number of seals after [tex]\( t \)[/tex] months. The general form of an exponential growth function is:
[tex]\[
y = y_0 \cdot (1 + r_{\text{monthly}})^t
\][/tex]
Here:
- [tex]\( y \)[/tex] is the number of seals after [tex]\( t \)[/tex] months,
- [tex]\( y_0 \)[/tex] is the initial population, which is 60,
- [tex]\( r_{\text{monthly}} \)[/tex] is the monthly growth rate, which we found to be [tex]\( 0.01171491691985338 \)[/tex].
Therefore, the exponential growth function is:
[tex]\[
y = 60 \cdot (1 + 0.01171491691985338)^t
\][/tex]
6. Summary:
- The monthly growth rate is approximately [tex]\( 0.01171491691985338 \)[/tex].
- The exponential growth function for the number of seals after [tex]\( t \)[/tex] months is:
[tex]\[
y = 60 \cdot (1 + 0.01171491691985338)^t
\][/tex]
This completes our detailed solution. If you have any further questions, feel free to ask!
