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Answer :
We are given the formula for exponential growth:
$$
P_t = P_0 \cdot 2^{\frac{t}{d}},
$$
where
- $P_0$ is the initial population,
- $t$ is the elapsed time in hours, and
- $d$ is the doubling time.
Step 1: Substitute the given values into the formula. Here, the initial population is $P_0 = 230$, the elapsed time is $t = 13$ hours, and the doubling time is $d = 9$ hours. Thus, we have
$$
P_{13} = 230 \cdot 2^{\frac{13}{9}}.
$$
Step 2: Compute the exponent $\frac{13}{9}$.
$$
\frac{13}{9} \approx 1.444444444.
$$
Step 3: Calculate the value of $2^{\frac{13}{9}}$. This yields a number approximately equal to
$$
2^{1.444444444} \approx 2.7220 \quad \text{(approximation)}.
$$
Step 4: Compute the product with the initial population:
$$
P_{13} \approx 230 \times 2.7220 \approx 625.9634.
$$
Step 5: Round the result to the nearest whole number:
$$
P_{13} \approx 626.
$$
Thus, after 13 hours, the population of bacteria in the culture is approximately
$$
\boxed{626}.
$$
$$
P_t = P_0 \cdot 2^{\frac{t}{d}},
$$
where
- $P_0$ is the initial population,
- $t$ is the elapsed time in hours, and
- $d$ is the doubling time.
Step 1: Substitute the given values into the formula. Here, the initial population is $P_0 = 230$, the elapsed time is $t = 13$ hours, and the doubling time is $d = 9$ hours. Thus, we have
$$
P_{13} = 230 \cdot 2^{\frac{13}{9}}.
$$
Step 2: Compute the exponent $\frac{13}{9}$.
$$
\frac{13}{9} \approx 1.444444444.
$$
Step 3: Calculate the value of $2^{\frac{13}{9}}$. This yields a number approximately equal to
$$
2^{1.444444444} \approx 2.7220 \quad \text{(approximation)}.
$$
Step 4: Compute the product with the initial population:
$$
P_{13} \approx 230 \times 2.7220 \approx 625.9634.
$$
Step 5: Round the result to the nearest whole number:
$$
P_{13} \approx 626.
$$
Thus, after 13 hours, the population of bacteria in the culture is approximately
$$
\boxed{626}.
$$
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