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Answer :
To find the population of bacteria in the culture after 11 hours, we'll use the formula:
[tex]\[ P_t = P_0 \cdot 2^{\frac{t}{d}} \][/tex]
Here’s a step-by-step explanation of how to use this formula:
1. Identify the Variables:
- [tex]\( P_0 \)[/tex]: the initial population of bacteria, which is 460.
- [tex]\( t \)[/tex]: the time in hours after the initial population, which is 11 hours.
- [tex]\( d \)[/tex]: the doubling time, which is 4 hours.
- [tex]\( P_t \)[/tex]: the population after [tex]\( t \)[/tex] hours, which is what we want to find.
2. Substitute the Values into the Formula:
- Replace [tex]\( P_0 \)[/tex] with 460, [tex]\( t \)[/tex] with 11, and [tex]\( d \)[/tex] with 4 in the formula.
[tex]\[
P_t = 460 \cdot 2^{\frac{11}{4}}
\][/tex]
3. Calculate the Exponent:
- First, compute [tex]\(\frac{11}{4}\)[/tex], which is 2.75.
- Then, calculate [tex]\(2^{2.75}\)[/tex].
4. Calculate [tex]\( P_t \)[/tex]:
- Multiply the initial population (460) by [tex]\(2^{2.75}\)[/tex] to find [tex]\( P_t \)[/tex].
5. Round the Result:
- The population after 11 hours is calculated to be approximately 3094 when rounded to the nearest whole number.
Therefore, the population of bacteria after 11 hours is approximately 3094.
[tex]\[ P_t = P_0 \cdot 2^{\frac{t}{d}} \][/tex]
Here’s a step-by-step explanation of how to use this formula:
1. Identify the Variables:
- [tex]\( P_0 \)[/tex]: the initial population of bacteria, which is 460.
- [tex]\( t \)[/tex]: the time in hours after the initial population, which is 11 hours.
- [tex]\( d \)[/tex]: the doubling time, which is 4 hours.
- [tex]\( P_t \)[/tex]: the population after [tex]\( t \)[/tex] hours, which is what we want to find.
2. Substitute the Values into the Formula:
- Replace [tex]\( P_0 \)[/tex] with 460, [tex]\( t \)[/tex] with 11, and [tex]\( d \)[/tex] with 4 in the formula.
[tex]\[
P_t = 460 \cdot 2^{\frac{11}{4}}
\][/tex]
3. Calculate the Exponent:
- First, compute [tex]\(\frac{11}{4}\)[/tex], which is 2.75.
- Then, calculate [tex]\(2^{2.75}\)[/tex].
4. Calculate [tex]\( P_t \)[/tex]:
- Multiply the initial population (460) by [tex]\(2^{2.75}\)[/tex] to find [tex]\( P_t \)[/tex].
5. Round the Result:
- The population after 11 hours is calculated to be approximately 3094 when rounded to the nearest whole number.
Therefore, the population of bacteria after 11 hours is approximately 3094.
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