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Answer :
To solve this problem, we will use the formula for exponential 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.
- [tex]\( t \)[/tex] is the time in hours.
- [tex]\( d \)[/tex] is the doubling time in hours.
Given:
- Initial population, [tex]\( P_0 = 4800 \)[/tex]
- Time, [tex]\( t = 14 \)[/tex] hours
- Doubling time, [tex]\( d = 9 \)[/tex] hours
Let's plug these values into the formula to find the population after 14 hours:
1. Calculate the exponent:
[tex]\[
\frac{t}{d} = \frac{14}{9} \approx 1.5556
\][/tex]
2. Substitute into the formula:
[tex]\[
P_t = 4800 \cdot 2^{1.5556}
\][/tex]
3. Calculate [tex]\( 2^{1.5556} \)[/tex]:
[tex]\[
2^{1.5556} \approx 2.944
\][/tex]
4. Calculate the population after 14 hours:
[tex]\[
P_t \approx 4800 \cdot 2.944 \approx 14109.45
\][/tex]
5. Round to the nearest whole number:
[tex]\[
P_t \approx 14109
\][/tex]
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.
- [tex]\( t \)[/tex] is the time in hours.
- [tex]\( d \)[/tex] is the doubling time in hours.
Given:
- Initial population, [tex]\( P_0 = 4800 \)[/tex]
- Time, [tex]\( t = 14 \)[/tex] hours
- Doubling time, [tex]\( d = 9 \)[/tex] hours
Let's plug these values into the formula to find the population after 14 hours:
1. Calculate the exponent:
[tex]\[
\frac{t}{d} = \frac{14}{9} \approx 1.5556
\][/tex]
2. Substitute into the formula:
[tex]\[
P_t = 4800 \cdot 2^{1.5556}
\][/tex]
3. Calculate [tex]\( 2^{1.5556} \)[/tex]:
[tex]\[
2^{1.5556} \approx 2.944
\][/tex]
4. Calculate the population after 14 hours:
[tex]\[
P_t \approx 4800 \cdot 2.944 \approx 14109.45
\][/tex]
5. Round to the nearest whole number:
[tex]\[
P_t \approx 14109
\][/tex]
So, the population of bacteria in the culture after 14 hours is approximately 14,109.
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