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Answer :
To solve this problem, let's go through the steps using the formula provided for the population of bacteria:
The formula is:
[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.
Now, let's plug the given values into the formula:
- The initial population ([tex]\( P_0 \)[/tex]) is 43,000 bacteria.
- The doubling time ([tex]\( d \)[/tex]) is 5 hours.
- The time ([tex]\( t \)[/tex]) after which we want to find the population is 13 hours.
Substitute these values into the formula:
[tex]\[ P_t = 43000 \cdot 2^{\frac{13}{5}} \][/tex]
1. First, calculate the exponent:
[tex]\[ \frac{13}{5} = 2.6 \][/tex]
2. Next, calculate [tex]\( 2^{2.6} \)[/tex]:
[tex]\[ 2^{2.6} \approx 6.209 \][/tex] (rounded approximately)
3. Now, multiply this result by the initial population:
[tex]\[ 43000 \times 6.209 \approx 267,703 \][/tex]
4. Finally, round to the nearest whole number:
[tex]\[ 267,703 \][/tex]
So, the population of bacteria in the culture after 13 hours is approximately 260,703.
The formula is:
[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.
Now, let's plug the given values into the formula:
- The initial population ([tex]\( P_0 \)[/tex]) is 43,000 bacteria.
- The doubling time ([tex]\( d \)[/tex]) is 5 hours.
- The time ([tex]\( t \)[/tex]) after which we want to find the population is 13 hours.
Substitute these values into the formula:
[tex]\[ P_t = 43000 \cdot 2^{\frac{13}{5}} \][/tex]
1. First, calculate the exponent:
[tex]\[ \frac{13}{5} = 2.6 \][/tex]
2. Next, calculate [tex]\( 2^{2.6} \)[/tex]:
[tex]\[ 2^{2.6} \approx 6.209 \][/tex] (rounded approximately)
3. Now, multiply this result by the initial population:
[tex]\[ 43000 \times 6.209 \approx 267,703 \][/tex]
4. Finally, round to the nearest whole number:
[tex]\[ 267,703 \][/tex]
So, the population of bacteria in the culture after 13 hours is approximately 260,703.
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