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Answer :
- Substitute the given values into the formula: $P_{13} = 43000 \cdot 2^{\frac{13}{5}}$.
- Calculate $2^{\frac{13}{5}} \approx 6.062866$.
- Multiply the result by 43000: $P_{13} = 43000 \times 6.062866 \approx 260703.238$.
- Round the final result to the nearest whole number: $P_{13} \approx \boxed{260703}$.
### Explanation
1. Understanding the Problem
We are given the formula for population growth: $P_t=P_0 "." 2^{\frac{t}{d}}$, 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:
- Initial population, $P_0 = 43000$
- Doubling time, $d = 5$ hours
- Time elapsed, $t = 13$ hours
We want to find the population of bacteria after 13 hours, $P_{13}$, rounded to the nearest whole number.
2. Substituting the Values
Substitute the given values into the formula:
$P_{13} = 43000 "." 2^{\frac{13}{5}}$
3. Calculating the Exponential Term
Calculate $2^{\frac{13}{5}}$:
$2^{\frac{13}{5}} \approx 6.062866$
4. Finding the Population
Multiply the result by 43000:
$P_{13} = 43000 \times 6.062866 \approx 260703.238$
5. Rounding to the Nearest Whole Number
Round the final result to the nearest whole number:
$P_{13} \approx 260703$
Therefore, the population of bacteria in the culture after 13 hours is approximately 260703.
### Examples
Understanding exponential growth is crucial in various real-world scenarios. For instance, it helps in predicting the spread of diseases, such as the flu or a pandemic, by modeling how quickly the number of infected individuals can increase over time. Similarly, in finance, it's used to calculate compound interest, showing how investments can grow exponentially over the years. In environmental science, it can model population growth or decay, aiding in conservation efforts and resource management.
- Calculate $2^{\frac{13}{5}} \approx 6.062866$.
- Multiply the result by 43000: $P_{13} = 43000 \times 6.062866 \approx 260703.238$.
- Round the final result to the nearest whole number: $P_{13} \approx \boxed{260703}$.
### Explanation
1. Understanding the Problem
We are given the formula for population growth: $P_t=P_0 "." 2^{\frac{t}{d}}$, 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:
- Initial population, $P_0 = 43000$
- Doubling time, $d = 5$ hours
- Time elapsed, $t = 13$ hours
We want to find the population of bacteria after 13 hours, $P_{13}$, rounded to the nearest whole number.
2. Substituting the Values
Substitute the given values into the formula:
$P_{13} = 43000 "." 2^{\frac{13}{5}}$
3. Calculating the Exponential Term
Calculate $2^{\frac{13}{5}}$:
$2^{\frac{13}{5}} \approx 6.062866$
4. Finding the Population
Multiply the result by 43000:
$P_{13} = 43000 \times 6.062866 \approx 260703.238$
5. Rounding to the Nearest Whole Number
Round the final result to the nearest whole number:
$P_{13} \approx 260703$
Therefore, the population of bacteria in the culture after 13 hours is approximately 260703.
### Examples
Understanding exponential growth is crucial in various real-world scenarios. For instance, it helps in predicting the spread of diseases, such as the flu or a pandemic, by modeling how quickly the number of infected individuals can increase over time. Similarly, in finance, it's used to calculate compound interest, showing how investments can grow exponentially over the years. In environmental science, it can model population growth or decay, aiding in conservation efforts and resource management.
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