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Answer :
To find the population of bacteria 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 in hours.
Let's go through the steps:
1. Identify the given values:
- Initial population, [tex]\( P_0 = 400 \)[/tex] bacteria.
- Doubling time, [tex]\( d = 2 \)[/tex] hours.
- Time, [tex]\( t = 13 \)[/tex] hours.
2. Substitute the values into the formula:
[tex]\[
P_t = 400 \cdot 2^{\frac{13}{2}}
\][/tex]
This means you will calculate the power of 2 using [tex]\( \frac{13}{2} \)[/tex], which is 6.5.
3. Calculate the power of 2:
First, determine [tex]\( 2^{6.5} \)[/tex]. This involves calculations that result in a number that represents how many times the population will double in those 13 hours.
4. Multiply by the initial population:
Multiply the result of [tex]\( 2^{6.5} \)[/tex] by 400 to get the population after 13 hours.
5. Round to the nearest whole number:
After computing the entire expression, round the result to the nearest whole number to find the final population count as bacteria count should always be a whole number.
Based on these calculations, the population of bacteria in the culture after 13 hours is approximately 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 in hours.
Let's go through the steps:
1. Identify the given values:
- Initial population, [tex]\( P_0 = 400 \)[/tex] bacteria.
- Doubling time, [tex]\( d = 2 \)[/tex] hours.
- Time, [tex]\( t = 13 \)[/tex] hours.
2. Substitute the values into the formula:
[tex]\[
P_t = 400 \cdot 2^{\frac{13}{2}}
\][/tex]
This means you will calculate the power of 2 using [tex]\( \frac{13}{2} \)[/tex], which is 6.5.
3. Calculate the power of 2:
First, determine [tex]\( 2^{6.5} \)[/tex]. This involves calculations that result in a number that represents how many times the population will double in those 13 hours.
4. Multiply by the initial population:
Multiply the result of [tex]\( 2^{6.5} \)[/tex] by 400 to get the population after 13 hours.
5. Round to the nearest whole number:
After computing the entire expression, round the result to the nearest whole number to find the final population count as bacteria count should always be a whole number.
Based on these calculations, the population of bacteria in the culture after 13 hours is approximately 36,204.
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