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Answer :
- Substitute the given values into the formula: $P_{13} = 65000 "." 2^{\frac{13}{2}}$.
- Calculate $2^{\frac{13}{2}} \approx 90.50966799$.
- Multiply the result by 65000: $65000 \times 90.50966799 \approx 5883128.41947$.
- Round the final result to the nearest whole number: $P_{13} \approx \boxed{5883128}$.
### Explanation
1. Understanding the Problem
We are given the formula for the population of bacteria at time $t$: $P_t = P_0 "." 2^{\frac{t}{d}}$, 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 the following values:
- Initial population, $P_0 = 65000$
- Doubling time, $d = 2$ hours
- Time, $t = 13$ hours
We want to find the population of bacteria after 13 hours, rounded to the nearest whole number.
2. Calculations
Substitute the given values into the formula:
$$P_{13} = 65000 "." 2^{\frac{13}{2}}$$
First, we calculate $2^{\frac{13}{2}}$:
$$2^{\frac{13}{2}} = 2^{6.5} \approx 90.50966799$$
Next, we multiply this result by the initial population:
$$P_{13} = 65000 \times 90.50966799 \approx 5883128.41947$$
3. Final Answer
Finally, we round the result to the nearest whole number:
$$P_{13} \approx 5883128$$
So, the population of bacteria after 13 hours is approximately 5,883,128.
### Examples
Understanding exponential growth, as modeled by the bacteria population, is crucial in various real-world scenarios. For instance, it helps in predicting the spread of infectious diseases, managing financial investments with compound interest, and even understanding population dynamics in ecological systems. By grasping these concepts, one can make informed decisions in public health, finance, and environmental conservation, highlighting the broad applicability of exponential models.
- Calculate $2^{\frac{13}{2}} \approx 90.50966799$.
- Multiply the result by 65000: $65000 \times 90.50966799 \approx 5883128.41947$.
- Round the final result to the nearest whole number: $P_{13} \approx \boxed{5883128}$.
### Explanation
1. Understanding the Problem
We are given the formula for the population of bacteria at time $t$: $P_t = P_0 "." 2^{\frac{t}{d}}$, 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 the following values:
- Initial population, $P_0 = 65000$
- Doubling time, $d = 2$ hours
- Time, $t = 13$ hours
We want to find the population of bacteria after 13 hours, rounded to the nearest whole number.
2. Calculations
Substitute the given values into the formula:
$$P_{13} = 65000 "." 2^{\frac{13}{2}}$$
First, we calculate $2^{\frac{13}{2}}$:
$$2^{\frac{13}{2}} = 2^{6.5} \approx 90.50966799$$
Next, we multiply this result by the initial population:
$$P_{13} = 65000 \times 90.50966799 \approx 5883128.41947$$
3. Final Answer
Finally, we round the result to the nearest whole number:
$$P_{13} \approx 5883128$$
So, the population of bacteria after 13 hours is approximately 5,883,128.
### Examples
Understanding exponential growth, as modeled by the bacteria population, is crucial in various real-world scenarios. For instance, it helps in predicting the spread of infectious diseases, managing financial investments with compound interest, and even understanding population dynamics in ecological systems. By grasping these concepts, one can make informed decisions in public health, finance, and environmental conservation, highlighting the broad applicability of exponential models.
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