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Answer :
To find the population of bacteria after 14 hours, we can use the formula for population growth:
[tex]\[ P_t = P_0 \cdot 2^{\frac{t}{d}} \][/tex]
Where:
- [tex]\( P_t \)[/tex] is the population after [tex]\( t \)[/tex] hours.
- [tex]\( P_0 \)[/tex] is the initial population, which is 4800.
- [tex]\( t \)[/tex] is the time in hours, given as 14 hours.
- [tex]\( d \)[/tex] is the doubling time, which is 9 hours.
Let's calculate the population step-by-step:
1. Identify the initial values:
- Initial population, [tex]\( P_0 = 4800 \)[/tex]
- Time, [tex]\( t = 14 \)[/tex] hours
- Doubling time, [tex]\( d = 9 \)[/tex] hours
2. Substitute these values into the formula:
[tex]\[ P_t = 4800 \cdot 2^{\frac{14}{9}} \][/tex]
3. Calculate the exponent:
[tex]\(\frac{14}{9}\)[/tex] means dividing 14 by 9, which results in approximately 1.5556.
4. Evaluate the expression [tex]\(2^{1.5556}\)[/tex]:
[tex]\(2^{1.5556}\)[/tex] is approximately 2.9396.
5. Multiply the initial population by this factor:
[tex]\( P_t = 4800 \times 2.9396 = 14108.48 \)[/tex]
6. Round to the nearest whole number:
The population after rounding is approximately 14109.
So, the population of bacteria in the culture after 14 hours is approximately 14,109.
[tex]\[ P_t = P_0 \cdot 2^{\frac{t}{d}} \][/tex]
Where:
- [tex]\( P_t \)[/tex] is the population after [tex]\( t \)[/tex] hours.
- [tex]\( P_0 \)[/tex] is the initial population, which is 4800.
- [tex]\( t \)[/tex] is the time in hours, given as 14 hours.
- [tex]\( d \)[/tex] is the doubling time, which is 9 hours.
Let's calculate the population step-by-step:
1. Identify the initial values:
- Initial population, [tex]\( P_0 = 4800 \)[/tex]
- Time, [tex]\( t = 14 \)[/tex] hours
- Doubling time, [tex]\( d = 9 \)[/tex] hours
2. Substitute these values into the formula:
[tex]\[ P_t = 4800 \cdot 2^{\frac{14}{9}} \][/tex]
3. Calculate the exponent:
[tex]\(\frac{14}{9}\)[/tex] means dividing 14 by 9, which results in approximately 1.5556.
4. Evaluate the expression [tex]\(2^{1.5556}\)[/tex]:
[tex]\(2^{1.5556}\)[/tex] is approximately 2.9396.
5. Multiply the initial population by this factor:
[tex]\( P_t = 4800 \times 2.9396 = 14108.48 \)[/tex]
6. Round to the nearest whole number:
The population after rounding is approximately 14109.
So, the population of bacteria in the culture after 14 hours is approximately 14,109.
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