Answer :
- Substitute the given values into the formula: $P_t = P_0 Imes 2^{\frac{t}{d}}$ with $P_0 = 230$, $t = 13$, and $d = 9$.
- Calculate $P_{13} = 230 Imes 2^{\frac{13}{9}}$.
- The result of the calculation is approximately 625.9634.
- Round the result to the nearest whole number, which gives the final answer: $\boxed{626}$.
### Explanation
1. Understanding the Problem
We are given the formula for exponential growth of a bacterial population: $P_t = P_0 Imes 2^{\frac{t}{d}}$, where:
- $P_t$ is the population after $t$ hours
- $P_0$ is the initial population
- $t$ is the time in hours
- $d$ is the doubling time
We are given the following values:
- Initial population, $P_0 = 230$
- Doubling time, $d = 9$ hours
- Time, $t = 13$ hours
We want to find the population $P_{13}$ after 13 hours.
2. Substituting the Values
Now, we substitute the given values into the formula:
$P_{13} = 230 Imes 2^{\frac{13}{9}}$
3. Calculating the Population
We calculate the value of $2^{\frac{13}{9}}$:
$2^{\frac{13}{9}} \approx 2.72158$
So, $P_{13} = 230 Imes 2.72158 \approx 625.9634$
4. Rounding to the Nearest Whole Number
Since we need to round the population to the nearest whole number, we have:
$P_{13} \approx 626$
5. Final Answer
Therefore, the population of bacteria in the culture after 13 hours is approximately 626.
### Examples
Exponential growth models, like the one used here, are crucial in various real-world scenarios. For instance, they help in predicting the spread of diseases, managing financial investments, and understanding population dynamics. In epidemiology, these models can forecast the number of infected individuals during an outbreak, aiding in resource allocation and intervention strategies. Similarly, in finance, they can estimate the growth of investments over time, helping investors make informed decisions. Understanding exponential growth is therefore essential in many fields for making accurate predictions and effective planning.
- Calculate $P_{13} = 230 Imes 2^{\frac{13}{9}}$.
- The result of the calculation is approximately 625.9634.
- Round the result to the nearest whole number, which gives the final answer: $\boxed{626}$.
### Explanation
1. Understanding the Problem
We are given the formula for exponential growth of a bacterial population: $P_t = P_0 Imes 2^{\frac{t}{d}}$, where:
- $P_t$ is the population after $t$ hours
- $P_0$ is the initial population
- $t$ is the time in hours
- $d$ is the doubling time
We are given the following values:
- Initial population, $P_0 = 230$
- Doubling time, $d = 9$ hours
- Time, $t = 13$ hours
We want to find the population $P_{13}$ after 13 hours.
2. Substituting the Values
Now, we substitute the given values into the formula:
$P_{13} = 230 Imes 2^{\frac{13}{9}}$
3. Calculating the Population
We calculate the value of $2^{\frac{13}{9}}$:
$2^{\frac{13}{9}} \approx 2.72158$
So, $P_{13} = 230 Imes 2.72158 \approx 625.9634$
4. Rounding to the Nearest Whole Number
Since we need to round the population to the nearest whole number, we have:
$P_{13} \approx 626$
5. Final Answer
Therefore, the population of bacteria in the culture after 13 hours is approximately 626.
### Examples
Exponential growth models, like the one used here, are crucial in various real-world scenarios. For instance, they help in predicting the spread of diseases, managing financial investments, and understanding population dynamics. In epidemiology, these models can forecast the number of infected individuals during an outbreak, aiding in resource allocation and intervention strategies. Similarly, in finance, they can estimate the growth of investments over time, helping investors make informed decisions. Understanding exponential growth is therefore essential in many fields for making accurate predictions and effective planning.