IDNLearn.com offers a user-friendly platform for finding and sharing knowledge. Our Q&A platform offers reliable and thorough answers to ensure you have the information you need to succeed in any situation.
Sagot :
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!
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!
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Your search for answers ends at IDNLearn.com. Thanks for visiting, and we look forward to helping you again soon.