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Answer :
- Substitute the given values into the formula: $P_{11} = 560
cdot 2^{\frac{11}{10}}$.
- Calculate the exponential term: $2^{\frac{11}{10}} \approx 2.1435$.
- Multiply by the initial population: $P_{11} \approx 560
cdot 2.1435 \approx 1200.386$.
- Round to the nearest whole number: $P_{11} \approx \boxed{1200}$.
### Explanation
1. Understanding the Problem
We are given the formula for population growth: $P_t = P_0
cdot 2^{\frac{t}{d}}$, 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, $P_0 = 560$
- Doubling time, $d = 10$ hours
- Time elapsed, $t = 11$ hours
We want to find the population of bacteria after 11 hours, $P_{11}$.
2. Substituting the Values
Substitute the given values into the formula:
$$P_{11} = 560
cdot 2^{\frac{11}{10}}$$
3. Calculating the Exponential Term
Calculate $2^{\frac{11}{10}}$:
$$2^{\frac{11}{10}} \approx 2^{1.1} \approx 2.1435$$
4. Finding the Population
Multiply the result by 560:
$$P_{11} = 560
cdot 2.1435 \approx 1200.386$$
5. Rounding to the Nearest Whole Number
Round $P_{11}$ to the nearest whole number:
$$P_{11} \approx 1200$$
Therefore, the population of bacteria in the culture after 11 hours is approximately 1200.
### 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 COVID-19, where the number of infected individuals can double within a specific time frame. Similarly, in finance, compound interest follows an exponential growth pattern, allowing investors to estimate how their investments will grow over time. This concept is also vital in environmental science, where it's used to model population growth or the decay of radioactive materials.
cdot 2^{\frac{11}{10}}$.
- Calculate the exponential term: $2^{\frac{11}{10}} \approx 2.1435$.
- Multiply by the initial population: $P_{11} \approx 560
cdot 2.1435 \approx 1200.386$.
- Round to the nearest whole number: $P_{11} \approx \boxed{1200}$.
### Explanation
1. Understanding the Problem
We are given the formula for population growth: $P_t = P_0
cdot 2^{\frac{t}{d}}$, 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, $P_0 = 560$
- Doubling time, $d = 10$ hours
- Time elapsed, $t = 11$ hours
We want to find the population of bacteria after 11 hours, $P_{11}$.
2. Substituting the Values
Substitute the given values into the formula:
$$P_{11} = 560
cdot 2^{\frac{11}{10}}$$
3. Calculating the Exponential Term
Calculate $2^{\frac{11}{10}}$:
$$2^{\frac{11}{10}} \approx 2^{1.1} \approx 2.1435$$
4. Finding the Population
Multiply the result by 560:
$$P_{11} = 560
cdot 2.1435 \approx 1200.386$$
5. Rounding to the Nearest Whole Number
Round $P_{11}$ to the nearest whole number:
$$P_{11} \approx 1200$$
Therefore, the population of bacteria in the culture after 11 hours is approximately 1200.
### 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 COVID-19, where the number of infected individuals can double within a specific time frame. Similarly, in finance, compound interest follows an exponential growth pattern, allowing investors to estimate how their investments will grow over time. This concept is also vital in environmental science, where it's used to model population growth or the decay of radioactive materials.
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