Answer :
To determine the expected population of deer in a region of a national park, we are given the following information:
- The initial population of deer is 200.
- The population grows at a rate of 7% per year.
From this information, we need to develop an exponential function that models the expected population over time. Here are the steps to solve this problem:
### Step-by-Step Solution:
1. Understanding the Growth Rate:
- A growth rate of 7% per year means that each year the population grows by 7% of its current size.
- An increase of 7% is the same as multiplying the population by 1.07 each year.
2. Exponential Growth Model:
- Exponential growth can be modeled by the function [tex]\( f(x) = a \cdot (b)^x \)[/tex], where:
- [tex]\( a \)[/tex] is the initial amount (starting population).
- [tex]\( b \)[/tex] is the growth factor (1 + growth rate in decimal form).
- [tex]\( x \)[/tex] is the number of time periods (years).
3. Determine the Components:
- The initial population ([tex]\( a \)[/tex]) is 200 deer.
- The growth factor ([tex]\( b \)[/tex]) is 1.07 (since 7% growth rate means each year's population is 107% of the previous year's population).
4. Construct the Function:
- Using the given values, the exponential function modeling the deer population [tex]\( f(x) \)[/tex] will be:
[tex]\[
f(x) = 200 \cdot (1.07)^x
\][/tex]
### Conclusion:
The exponential function that models the expected population of deer in the national park is:
[tex]\[ f(x) = 200 \cdot (1.07)^x \][/tex]
This function will provide the number of deer expected after [tex]\( x \)[/tex] years.
- The initial population of deer is 200.
- The population grows at a rate of 7% per year.
From this information, we need to develop an exponential function that models the expected population over time. Here are the steps to solve this problem:
### Step-by-Step Solution:
1. Understanding the Growth Rate:
- A growth rate of 7% per year means that each year the population grows by 7% of its current size.
- An increase of 7% is the same as multiplying the population by 1.07 each year.
2. Exponential Growth Model:
- Exponential growth can be modeled by the function [tex]\( f(x) = a \cdot (b)^x \)[/tex], where:
- [tex]\( a \)[/tex] is the initial amount (starting population).
- [tex]\( b \)[/tex] is the growth factor (1 + growth rate in decimal form).
- [tex]\( x \)[/tex] is the number of time periods (years).
3. Determine the Components:
- The initial population ([tex]\( a \)[/tex]) is 200 deer.
- The growth factor ([tex]\( b \)[/tex]) is 1.07 (since 7% growth rate means each year's population is 107% of the previous year's population).
4. Construct the Function:
- Using the given values, the exponential function modeling the deer population [tex]\( f(x) \)[/tex] will be:
[tex]\[
f(x) = 200 \cdot (1.07)^x
\][/tex]
### Conclusion:
The exponential function that models the expected population of deer in the national park is:
[tex]\[ f(x) = 200 \cdot (1.07)^x \][/tex]
This function will provide the number of deer expected after [tex]\( x \)[/tex] years.
