To solve the problem, let's break it down into its two parts: evaluating [tex]\( P(Q) \)[/tex] and predicting the population of Los Angeles on January 1, 2016.
### Part a: Evaluate [tex]\( P(Q) \)[/tex]
First, we need to understand the given function:
[tex]\[
P(t) = \frac{3.194}{1 + 14.589 e^{0.02 \pi t}}
\][/tex]
Here, [tex]\( t \)[/tex] represents the number of years since 1900. To evaluate [tex]\( P(Q) \)[/tex], we assume [tex]\( Q \)[/tex] is a particular value of [tex]\( t \)[/tex].
If we substitute [tex]\( t = Q \)[/tex], we get:
[tex]\[
P(Q) = \frac{3.194}{1 + 14.589 e^{0.02 \pi Q}}
\][/tex]
For the sake of evaluation, let's consider a specific scenario where [tex]\( Q = 0 \)[/tex] (i.e., the year 1900):
[tex]\[
P(Q) = \frac{3.194}{1 + 14.589 e^{0.02 \pi \cdot 0}}
\][/tex]
Since [tex]\( e^{0} = 1 \)[/tex]:
[tex]\[
P(Q) = \frac{3.194}{1 + 14.589 \cdot 1} = \frac{3.194}{15.589}
\][/tex]
Evaluating this:
[tex]\[
P(Q) \approx 0.20488806209506702 \text{ million}
\][/tex]
In the context of the problem, [tex]\( P(Q) \)[/tex] tells us the population of Los Angeles at a specific time [tex]\( Q \)[/tex]. For [tex]\( Q = 0 \)[/tex], it indicates the population in the year 1900.
### Part b: Predicting the population on January 1, 2016
To predict the population in 2016, we first find [tex]\( t \)[/tex] for the year 2016:
[tex]\[
t = 2016 - 1900 = 116
\][/tex]
Now, substitute [tex]\( t = 116 \)[/tex] into the logistic growth function:
[tex]\[
P(116) = \frac{3.194}{1 + 14.589 e^{0.02 \pi \cdot 116}}
\][/tex]
Evaluating the exponent first:
[tex]\[
0.02 \pi \cdot 116 \approx 7.27776
\][/tex]
Thus the function becomes:
[tex]\[
P(116) = \frac{3.194}{1 + 14.589 e^{7.27776}}
\][/tex]
Given [tex]\( e^{7.27776} \)[/tex] is a very large number, the denominator will be significantly larger than 1, making the fraction very small. Calculating this gives us:
[tex]\[
P(116) \approx 0.00014960148499918073 \text{ million}
\][/tex]
In the context of the problem, [tex]\( P(116) \)[/tex] represents the predicted population of Los Angeles on January 1, 2016. The result of approximately 0.00015 million suggests an extremely small population, likely indicating that the exponential growth term [tex]\( e^{0.02 \pi t} \)[/tex] vastly overshoots in this model, potentially due to limitations of the logistic growth formulation for large [tex]\( t \)[/tex].
