Let's solve the problem step-by-step:
Given the logistic growth function:
[tex]\[ f(t) = \frac{360}{1 + 11.0 e^{-0.12 t}} \][/tex]
### Part (a): Initial number of butterflies ([tex]\( t=0 \)[/tex])
To find the initial number of butterflies, we need to evaluate the function at [tex]\( t = 0 \)[/tex].
[tex]\[ f(0) = \frac{360}{1 + 11.0 e^{-0.12 \cdot 0}} \][/tex]
Since [tex]\( e^0 = 1 \)[/tex]:
[tex]\[ f(0) = \frac{360}{1 + 11.0 \cdot 1} \][/tex]
[tex]\[ f(0) = \frac{360}{12} \][/tex]
[tex]\[ f(0) = 30 \][/tex]
So, the initial number of butterflies is 30 butterflies.
### Part (b): Maximum (limiting) number of butterflies ([tex]\( t \to \infty \)[/tex])
As [tex]\( t \)[/tex] approaches infinity, the exponential term [tex]\( e^{-0.12 t} \)[/tex] approaches zero.
[tex]\[ \lim_{{t \to \infty}} f(t) = \frac{360}{1 + 11.0 \cdot 0} = \frac{360}{1} = 360 \][/tex]
So, the maximum (limiting) number of butterflies is 360 butterflies.
### Part (c): Number of butterflies expected after 20 months ([tex]\( t = 20 \)[/tex])
To find the number of butterflies after 20 months, we need to evaluate the function at [tex]\( t = 20 \)[/tex].
[tex]\[ f(20) = \frac{360}{1 + 11.0 e^{-0.12 \cdot 20}} \][/tex]
First, calculate the exponent:
[tex]\[ \text{Exponent} = -0.12 \cdot 20 = -2.4 \][/tex]
Then, evaluate the exponential term:
[tex]\[ e^{-2.4} \approx 0.0907 \][/tex]
Now, substitute this back into the function:
[tex]\[ f(20) = \frac{360}{1 + 11.0 \cdot 0.0907} \][/tex]
[tex]\[ f(20) = \frac{360}{1 + 0.9977} \][/tex]
[tex]\[ f(20) = \frac{360}{1.9977} \][/tex]
[tex]\[ f(20) \approx 180.19 \][/tex]
So, the number of butterflies expected in the habitat after 20 months is approximately 180 butterflies.
### Summary:
a) The initial number of butterflies is 30 butterflies.
b) The maximum (limiting) number of butterflies is 360 butterflies.
c) The number of butterflies expected after 20 months is approximately 180 butterflies.
