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Answer :
To find the population of bacteria in the culture after 13 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,
- [tex]\( t \)[/tex] is the time in hours,
- [tex]\( d \)[/tex] is the doubling time, which is 2 hours in this case.
Here is a step-by-step breakdown of the solution:
1. Identify the given values:
- Initial population ([tex]\( P_0 \)[/tex]) = 400 bacteria
- Doubling time ([tex]\( d \)[/tex]) = 2 hours
- Time elapsed ([tex]\( t \)[/tex]) = 13 hours
2. Substitute the values into the formula:
[tex]\[ P_t = 400 \cdot 2^{\frac{13}{2}} \][/tex]
3. Calculate the exponent:
- First, calculate [tex]\( \frac{13}{2} \)[/tex], which equals 6.5.
4. Calculate [tex]\( 2^{6.5} \)[/tex]:
- This involves finding the value of 2 raised to the power of 6.5.
5. Multiply by the initial population (400):
- Once you have [tex]\( 2^{6.5} \)[/tex], multiply it by 400 to find [tex]\( P_t \)[/tex].
6. Round the result to the nearest whole number:
- After performing these calculations, the population of bacteria in the culture after 13 hours is approximately 36,204 bacteria.
So, the population of bacteria after 13 hours is 36,204.
[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, which is 2 hours in this case.
Here is a step-by-step breakdown of the solution:
1. Identify the given values:
- Initial population ([tex]\( P_0 \)[/tex]) = 400 bacteria
- Doubling time ([tex]\( d \)[/tex]) = 2 hours
- Time elapsed ([tex]\( t \)[/tex]) = 13 hours
2. Substitute the values into the formula:
[tex]\[ P_t = 400 \cdot 2^{\frac{13}{2}} \][/tex]
3. Calculate the exponent:
- First, calculate [tex]\( \frac{13}{2} \)[/tex], which equals 6.5.
4. Calculate [tex]\( 2^{6.5} \)[/tex]:
- This involves finding the value of 2 raised to the power of 6.5.
5. Multiply by the initial population (400):
- Once you have [tex]\( 2^{6.5} \)[/tex], multiply it by 400 to find [tex]\( P_t \)[/tex].
6. Round the result to the nearest whole number:
- After performing these calculations, the population of bacteria in the culture after 13 hours is approximately 36,204 bacteria.
So, the population of bacteria after 13 hours is 36,204.
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