Let's analyze and solve the problem step-by-step using the given continuous growth model for the population of Pokémon in a nature preserve. The growth model is given by the equation:
[tex]\[ P = f(t) = 8600 e^{0.057t} \][/tex]
where [tex]\( P \)[/tex] is the number of Pokémon and [tex]\( t \)[/tex] is in years.
### Part 1: Finding [tex]\( f(0) \)[/tex]
First, we need to find the initial population of Pokémon when [tex]\( t = 0 \)[/tex].
[tex]\[ f(0) = 8600 e^{0} \][/tex]
Since [tex]\( e^0 \)[/tex] is 1, we have:
[tex]\[ f(0) = 8600 \times 1 = 8600 \][/tex]
Therefore, [tex]\( f(0) = 8600 \)[/tex].
### Part 2: Number of Pokémon after 7 years
Next, we need to determine the number of Pokémon after 7 years. To do this, we substitute [tex]\( t = 7 \)[/tex] into the growth model equation:
[tex]\[ P(7) = 8600 e^{0.057 \times 7} \][/tex]
Using the provided numerical results, the calculated number of Pokémon after 7 years is approximately 12817.
So, the number of Pokémon after 7 years, rounded to the nearest whole number, is:
[tex]\[ \boxed{12817} \][/tex]
### Part 3: Time to reach a population of 9800 Pokémon
Finally, let's determine how long it will take for the Pokémon population to grow to 9800. We set [tex]\( P = 9800 \)[/tex] and solve for [tex]\( t \)[/tex] in the equation:
[tex]\[ 9800 = 8600 e^{0.057t} \][/tex]
First, we isolate the exponential term by dividing both sides by 8600:
[tex]\[ \frac{9800}{8600} = e^{0.057t} \][/tex]
Next, simplify the fraction:
[tex]\[ \frac{9800}{8600} \approx 1.1395 \][/tex]
Now, we take the natural logarithm of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[ \ln(1.1395) = 0.057t \][/tex]
Then solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{\ln(1.1395)}{0.057} \][/tex]
Based on the numerical result, the time needed for the population to reach 9800 Pokémon is approximately 2.3 years.
So, the time it will take for the population to grow to 9800 Pokémon, rounded to the nearest tenth of a year, is:
[tex]\[ \boxed{2.3} \][/tex] years
