Answer :
To solve the problem of finding the bacterial population after a certain amount of time, 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.
Given:
- Initial population, [tex]\( P_0 = 46000 \)[/tex] bacteria
- Time, [tex]\( t = 11 \)[/tex] hours
- Doubling time, [tex]\( d = 6 \)[/tex] hours
Steps:
1. Calculate the exponent:
The exponent in the formula is [tex]\(\frac{t}{d}\)[/tex], which tells us how many times the population will have doubled. Here, it is [tex]\(\frac{11}{6}\)[/tex].
2. Exponential growth calculation:
Substitute [tex]\(\frac{11}{6}\)[/tex] into the formula:
[tex]\[ 2^{\frac{11}{6}} \][/tex]
This calculates how many times the population has doubled in this time period.
3. Determine new population:
Multiply the initial population by the doubling factor:
[tex]\[ P_t = 46000 \cdot 2^{\frac{11}{6}} \][/tex]
4. Final calculation:
Compute the result to find out the population after 11 hours, which turns out to be approximately 163,925 bacteria when rounded to the nearest whole number.
Therefore, the bacterial population after 11 hours is 163,925.
[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.
Given:
- Initial population, [tex]\( P_0 = 46000 \)[/tex] bacteria
- Time, [tex]\( t = 11 \)[/tex] hours
- Doubling time, [tex]\( d = 6 \)[/tex] hours
Steps:
1. Calculate the exponent:
The exponent in the formula is [tex]\(\frac{t}{d}\)[/tex], which tells us how many times the population will have doubled. Here, it is [tex]\(\frac{11}{6}\)[/tex].
2. Exponential growth calculation:
Substitute [tex]\(\frac{11}{6}\)[/tex] into the formula:
[tex]\[ 2^{\frac{11}{6}} \][/tex]
This calculates how many times the population has doubled in this time period.
3. Determine new population:
Multiply the initial population by the doubling factor:
[tex]\[ P_t = 46000 \cdot 2^{\frac{11}{6}} \][/tex]
4. Final calculation:
Compute the result to find out the population after 11 hours, which turns out to be approximately 163,925 bacteria when rounded to the nearest whole number.
Therefore, the bacterial population after 11 hours is 163,925.
