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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]\( P_0 \)[/tex] is the initial population, which is 43,000 bacteria.
- [tex]\( t \)[/tex] is the time in hours, which is 13 hours in this case.
- [tex]\( d \)[/tex] is the doubling time, which is 5 hours.
Let's break it down step-by-step:
1. Identify the initial values:
- Initial population [tex]\( P_0 = 43,000 \)[/tex]
- Time [tex]\( t = 13 \)[/tex] hours
- Doubling time [tex]\( d = 5 \)[/tex] hours
2. Plug the values into the formula:
[tex]\[ P_t = 43,000 \cdot 2^{\frac{13}{5}} \][/tex]
3. Calculate the exponent:
[tex]\[ \frac{13}{5} = 2.6 \][/tex]
4. Calculate [tex]\( 2^{2.6} \)[/tex]:
- [tex]\( 2^{2.6} \)[/tex] is approximately equal to 6.035
5. Now calculate [tex]\( P_t \)[/tex]:
[tex]\[ P_t = 43,000 \cdot 6.035 \][/tex]
[tex]\[ P_t \approx 260,703.25 \][/tex]
6. Round to the nearest whole number:
The population, when rounded, is approximately 260,703.
Therefore, the population of bacteria after 13 hours is approximately 260,703 bacteria.
[tex]\[ P_t = P_0 \cdot 2^{\frac{t}{d}} \][/tex]
where:
- [tex]\( P_0 \)[/tex] is the initial population, which is 43,000 bacteria.
- [tex]\( t \)[/tex] is the time in hours, which is 13 hours in this case.
- [tex]\( d \)[/tex] is the doubling time, which is 5 hours.
Let's break it down step-by-step:
1. Identify the initial values:
- Initial population [tex]\( P_0 = 43,000 \)[/tex]
- Time [tex]\( t = 13 \)[/tex] hours
- Doubling time [tex]\( d = 5 \)[/tex] hours
2. Plug the values into the formula:
[tex]\[ P_t = 43,000 \cdot 2^{\frac{13}{5}} \][/tex]
3. Calculate the exponent:
[tex]\[ \frac{13}{5} = 2.6 \][/tex]
4. Calculate [tex]\( 2^{2.6} \)[/tex]:
- [tex]\( 2^{2.6} \)[/tex] is approximately equal to 6.035
5. Now calculate [tex]\( P_t \)[/tex]:
[tex]\[ P_t = 43,000 \cdot 6.035 \][/tex]
[tex]\[ P_t \approx 260,703.25 \][/tex]
6. Round to the nearest whole number:
The population, when rounded, is approximately 260,703.
Therefore, the population of bacteria after 13 hours is approximately 260,703 bacteria.
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