Answer :
To find the population of bacteria after 13 hours, we can use the formula for exponential growth of a doubling population:
[tex]\[ P_t = P_0 \times 2^{\frac{t}{d}} \][/tex]
Here:
- [tex]\( P_t \)[/tex] is the population after time [tex]\( t \)[/tex],
- [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.
Given:
- Initial population ([tex]\( P_0 \)[/tex]) = 65,000 bacteria,
- Doubling time ([tex]\( d \)[/tex]) = 2 hours,
- Time ([tex]\( t \)[/tex]) = 13 hours.
Let's plug these values into the formula:
1. Calculate the exponent: The time in hours is divided by the doubling time. Here, it's [tex]\( \frac{13}{2} = 6.5 \)[/tex].
2. Apply the formula:
[tex]\[ P_t = 65,000 \times 2^{6.5} \][/tex]
3. Calculate [tex]\( 2^{6.5} \)[/tex]:
This value is a little bit above 2 raised to the 6th power, which is 64. In this step, you need to compute the precise value of [tex]\( 2^{6.5} \)[/tex], which is approximately 90.51.
4. Find the population:
[tex]\[ P_t = 65,000 \times 90.51 \][/tex]
[tex]\[ P_t \approx 5,883,128.42 \][/tex]
5. Round to the nearest whole number: The result should be rounded to the nearest whole number, which gives approximately 5,883,128.
Therefore, the population of bacteria in the culture after 13 hours is approximately 5,883,128 bacteria.
[tex]\[ P_t = P_0 \times 2^{\frac{t}{d}} \][/tex]
Here:
- [tex]\( P_t \)[/tex] is the population after time [tex]\( t \)[/tex],
- [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.
Given:
- Initial population ([tex]\( P_0 \)[/tex]) = 65,000 bacteria,
- Doubling time ([tex]\( d \)[/tex]) = 2 hours,
- Time ([tex]\( t \)[/tex]) = 13 hours.
Let's plug these values into the formula:
1. Calculate the exponent: The time in hours is divided by the doubling time. Here, it's [tex]\( \frac{13}{2} = 6.5 \)[/tex].
2. Apply the formula:
[tex]\[ P_t = 65,000 \times 2^{6.5} \][/tex]
3. Calculate [tex]\( 2^{6.5} \)[/tex]:
This value is a little bit above 2 raised to the 6th power, which is 64. In this step, you need to compute the precise value of [tex]\( 2^{6.5} \)[/tex], which is approximately 90.51.
4. Find the population:
[tex]\[ P_t = 65,000 \times 90.51 \][/tex]
[tex]\[ P_t \approx 5,883,128.42 \][/tex]
5. Round to the nearest whole number: The result should be rounded to the nearest whole number, which gives approximately 5,883,128.
Therefore, the population of bacteria in the culture after 13 hours is approximately 5,883,128 bacteria.