Answer :
To find the equation that models the number of butterflies in the park after [tex]\( n \)[/tex] years, we need to follow these steps:
1. Determine the Initial Number of Butterflies:
The current number of butterflies is estimated to be [tex]\( 20 \)[/tex] thousand. This is our initial quantity, denoted as [tex]\( N_0 \)[/tex].
2. Identify the Growth Rate:
The population is expected to grow by [tex]\( 4 \% \)[/tex] per year. This growth rate can be expressed as a decimal by dividing 4 by 100, which gives us [tex]\( 0.04 \)[/tex].
3. Formulate the Population Growth Equation:
The general formula for exponential growth is:
[tex]\[ N(n) = N_0 \times (1 + r)^n \][/tex]
where:
- [tex]\( N(n) \)[/tex] is the number of butterflies after [tex]\( n \)[/tex] years,
- [tex]\( N_0 \)[/tex] is the initial number of butterflies,
- [tex]\( r \)[/tex] is the growth rate,
- [tex]\( n \)[/tex] is the number of years.
4. Substitute the Known Values into the Equation:
Given [tex]\( N_0 = 20 \)[/tex] (thousand butterflies) and [tex]\( r = 0.04 \)[/tex], we substitute these values into the formula:
[tex]\[ N(n) = 20 \times (1 + 0.04)^n \][/tex]
Simplifying the expression inside the parentheses:
[tex]\[ N(n) = 20 \times 1.04^n \][/tex]
Thus, the number of butterflies in thousands after [tex]\( n \)[/tex] years is modeled by the equation:
[tex]\[ N(n) = 20 \times 1.04^n \][/tex]
This equation will help scientists predict the butterfly population in the park as time progresses, taking into account the estimated annual growth rate.
1. Determine the Initial Number of Butterflies:
The current number of butterflies is estimated to be [tex]\( 20 \)[/tex] thousand. This is our initial quantity, denoted as [tex]\( N_0 \)[/tex].
2. Identify the Growth Rate:
The population is expected to grow by [tex]\( 4 \% \)[/tex] per year. This growth rate can be expressed as a decimal by dividing 4 by 100, which gives us [tex]\( 0.04 \)[/tex].
3. Formulate the Population Growth Equation:
The general formula for exponential growth is:
[tex]\[ N(n) = N_0 \times (1 + r)^n \][/tex]
where:
- [tex]\( N(n) \)[/tex] is the number of butterflies after [tex]\( n \)[/tex] years,
- [tex]\( N_0 \)[/tex] is the initial number of butterflies,
- [tex]\( r \)[/tex] is the growth rate,
- [tex]\( n \)[/tex] is the number of years.
4. Substitute the Known Values into the Equation:
Given [tex]\( N_0 = 20 \)[/tex] (thousand butterflies) and [tex]\( r = 0.04 \)[/tex], we substitute these values into the formula:
[tex]\[ N(n) = 20 \times (1 + 0.04)^n \][/tex]
Simplifying the expression inside the parentheses:
[tex]\[ N(n) = 20 \times 1.04^n \][/tex]
Thus, the number of butterflies in thousands after [tex]\( n \)[/tex] years is modeled by the equation:
[tex]\[ N(n) = 20 \times 1.04^n \][/tex]
This equation will help scientists predict the butterfly population in the park as time progresses, taking into account the estimated annual growth rate.
