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Answer :
To find the population of bacteria after 14 hours, we can use the formula:
[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 790 bacteria in this case.
- [tex]\( t \)[/tex] is the time in hours, which is 14 hours.
- [tex]\( d \)[/tex] is the doubling time in hours, which is 5 hours.
Let's follow the steps to calculate the population:
1. Identify the Initial Population and Doubling Time:
- Initial population ([tex]\( P_0 \)[/tex]): 790 bacteria
- Doubling time ([tex]\( d \)[/tex]): 5 hours
2. Determine the Time [tex]\( t \)[/tex]:
- We are interested in the population after 14 hours.
3. Substitute the Values into the Formula:
- Substitute [tex]\( P_0 = 790 \)[/tex], [tex]\( t = 14 \)[/tex], and [tex]\( d = 5 \)[/tex] into the formula:
[tex]\[ P_t = 790 \cdot 2^{\frac{14}{5}} \][/tex]
4. Calculate the Exponent:
- Calculate [tex]\( \frac{14}{5} \)[/tex], which gives us 2.8.
5. Calculate [tex]\( 2^{2.8} \)[/tex]:
- Compute [tex]\( 2^{2.8} \)[/tex], which is approximately 6.964.
6. Calculate the Final Population:
- Multiply the initial population by the result of the exponential calculation:
[tex]\[ P_t = 790 \cdot 6.964 \][/tex]
7. Round to the Nearest Whole Number:
- The calculation results in approximately 5501.96.
- Rounded to the nearest whole number, the population after 14 hours is 5502 bacteria.
Therefore, the population of bacteria in the culture after 14 hours is 5502.
[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 790 bacteria in this case.
- [tex]\( t \)[/tex] is the time in hours, which is 14 hours.
- [tex]\( d \)[/tex] is the doubling time in hours, which is 5 hours.
Let's follow the steps to calculate the population:
1. Identify the Initial Population and Doubling Time:
- Initial population ([tex]\( P_0 \)[/tex]): 790 bacteria
- Doubling time ([tex]\( d \)[/tex]): 5 hours
2. Determine the Time [tex]\( t \)[/tex]:
- We are interested in the population after 14 hours.
3. Substitute the Values into the Formula:
- Substitute [tex]\( P_0 = 790 \)[/tex], [tex]\( t = 14 \)[/tex], and [tex]\( d = 5 \)[/tex] into the formula:
[tex]\[ P_t = 790 \cdot 2^{\frac{14}{5}} \][/tex]
4. Calculate the Exponent:
- Calculate [tex]\( \frac{14}{5} \)[/tex], which gives us 2.8.
5. Calculate [tex]\( 2^{2.8} \)[/tex]:
- Compute [tex]\( 2^{2.8} \)[/tex], which is approximately 6.964.
6. Calculate the Final Population:
- Multiply the initial population by the result of the exponential calculation:
[tex]\[ P_t = 790 \cdot 6.964 \][/tex]
7. Round to the Nearest Whole Number:
- The calculation results in approximately 5501.96.
- Rounded to the nearest whole number, the population after 14 hours is 5502 bacteria.
Therefore, the population of bacteria in the culture after 14 hours is 5502.
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