To find the expression that calculates the population of a species after [tex]\( t \)[/tex] years, given that the population doubles every nine years and that the initial population is 100 individuals, we need to follow these steps:
1. Understand the Growth Pattern: The population doubles every 9 years. This means that if you know the initial population ([tex]\( P_0 \)[/tex]), the population after 9 years will be [tex]\( 2 \times P_0 \)[/tex]. After 18 years (which is 2 periods of 9 years each), it will be [tex]\( 4 \times P_0 \)[/tex] (since [tex]\( 2 \times 2 \times P_0 = 4 \times P_0 \)[/tex]), and so on.
2. General Formula: We generally use the formula for exponential growth, which is:
[tex]\[
P(t) = P_0 \times 2^{\frac{t}{T}}
\][/tex]
where [tex]\( P(t) \)[/tex] is the population at time [tex]\( t \)[/tex], [tex]\( P_0 \)[/tex] is the initial population, and [tex]\( T \)[/tex] is the period it takes for the population to double (in this case, [tex]\( T = 9 \)[/tex] years).
3. Substitute Known Values: Here, [tex]\( P_0 = 100 \)[/tex] and [tex]\( T = 9 \)[/tex]. Thus, the formula becomes:
[tex]\[
P(t) = 100 \times 2^{\frac{t}{9}}
\][/tex]
To confirm our understanding, let's check the given options:
1. [tex]\[ 2 \times 100^{9 t} \][/tex]
- This option does not correctly represent an exponential growth scenario where the base should be 2.
2. [tex]\[ 2 \times 100^{\frac{t}{9}} \][/tex]
- This also does not correctly represent the population growth formula relevant to this scenario.
3. [tex]\[ 100 \times 2^{9 t} \][/tex]
- This option incorrectly uses [tex]\( 9t \)[/tex] in the exponential factor, which results in a much faster growth rate than the actual doubling period of 9 years.
4. [tex]\[ 100 \times 2^{\frac{t}{9}} \][/tex]
- This option correctly applies the doubling factor with the right exponent, indicating that the population doubles every 9 years.
Thus, the correct expression to calculate the population [tex]\( t \)[/tex] years after the start is:
[tex]\[ 100 \times 2^{\frac{t}{9}} \][/tex]
So, the correct answer is:
[tex]\[ 100 \times 2^{\frac{t}{9}} \][/tex]
