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Answer :
Sure! Let's walk through the steps to find the population of bacteria after 13 hours.
1. Understand the formula:
- The formula we are using is: [tex]\( P_t = P_0 \cdot 2^{\frac{t}{d}} \)[/tex].
- Here, [tex]\( P_t \)[/tex] is the population after [tex]\( t \)[/tex] hours.
- [tex]\( P_0 \)[/tex] is the initial population, which is 400 bacteria.
- [tex]\( t \)[/tex] is the time in hours, which is 13.
- [tex]\( d \)[/tex] is the doubling time, which is 2 hours (the population doubles every 2 hours).
2. Plug in the values:
- [tex]\( P_0 = 400 \)[/tex]
- [tex]\( t = 13 \)[/tex]
- [tex]\( d = 2 \)[/tex]
3. Calculate the exponent:
- Find [tex]\(\frac{t}{d} = \frac{13}{2}\)[/tex].
4. Compute the power of 2:
- Calculate [tex]\(2^{\frac{13}{2}}\)[/tex].
5. Calculate [tex]\( P_t \)[/tex]:
- Multiply the initial population by the result from step 4: [tex]\( P_t = 400 \times 2^{6.5} \)[/tex].
6. Round to the nearest whole number:
- After computing, round the solution to the nearest whole number to get the final population.
Following these steps, the population of bacteria in the culture after 13 hours is 36,204.
1. Understand the formula:
- The formula we are using is: [tex]\( P_t = P_0 \cdot 2^{\frac{t}{d}} \)[/tex].
- Here, [tex]\( P_t \)[/tex] is the population after [tex]\( t \)[/tex] hours.
- [tex]\( P_0 \)[/tex] is the initial population, which is 400 bacteria.
- [tex]\( t \)[/tex] is the time in hours, which is 13.
- [tex]\( d \)[/tex] is the doubling time, which is 2 hours (the population doubles every 2 hours).
2. Plug in the values:
- [tex]\( P_0 = 400 \)[/tex]
- [tex]\( t = 13 \)[/tex]
- [tex]\( d = 2 \)[/tex]
3. Calculate the exponent:
- Find [tex]\(\frac{t}{d} = \frac{13}{2}\)[/tex].
4. Compute the power of 2:
- Calculate [tex]\(2^{\frac{13}{2}}\)[/tex].
5. Calculate [tex]\( P_t \)[/tex]:
- Multiply the initial population by the result from step 4: [tex]\( P_t = 400 \times 2^{6.5} \)[/tex].
6. Round to the nearest whole number:
- After computing, round the solution to the nearest whole number to get the final population.
Following these steps, the population of bacteria in the culture after 13 hours is 36,204.
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