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Answer :
To find the population of bacteria after 13 hours, we can use the formula for exponential growth:
[tex]\[ P_t = P_0 \cdot 2^{\frac{t}{d}} \][/tex]
Here's a step-by-step breakdown:
1. Identify the given values:
- Initial population, [tex]\( P_0 = 43,000 \)[/tex] bacteria
- 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:
- First, divide the time by the doubling time: [tex]\( \frac{13}{5} = 2.6 \)[/tex]
4. Calculate [tex]\( 2^{2.6} \)[/tex]:
- This represents how many times the population doubles in the given time.
5. Multiply by the initial population:
- Then, multiply this result by the initial population to find the bacteria population at 13 hours.
6. Round to the nearest whole number:
- Since we need the population as a whole number, rounding will give us the final answer.
Following these steps, the population of bacteria in the culture after 13 hours is approximately 260,703 bacteria.
[tex]\[ P_t = P_0 \cdot 2^{\frac{t}{d}} \][/tex]
Here's a step-by-step breakdown:
1. Identify the given values:
- Initial population, [tex]\( P_0 = 43,000 \)[/tex] bacteria
- 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:
- First, divide the time by the doubling time: [tex]\( \frac{13}{5} = 2.6 \)[/tex]
4. Calculate [tex]\( 2^{2.6} \)[/tex]:
- This represents how many times the population doubles in the given time.
5. Multiply by the initial population:
- Then, multiply this result by the initial population to find the bacteria population at 13 hours.
6. Round to the nearest whole number:
- Since we need the population as a whole number, rounding will give us the final answer.
Following these steps, the population of bacteria in the culture after 13 hours is approximately 260,703 bacteria.
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