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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]\( P_t \)[/tex] is the population after [tex]\( t \)[/tex] hours.
- [tex]\( P_0 \)[/tex] is the initial population, which is 610 bacteria.
- [tex]\( t \)[/tex] is the time in hours, which is 13 in this case.
- [tex]\( d \)[/tex] is the doubling time, which is 6 hours.
Let's go through the steps:
1. Identify the known values:
- Initial population, [tex]\( P_0 = 610 \)[/tex].
- Doubling time, [tex]\( d = 6 \)[/tex] hours.
- Time, [tex]\( t = 13 \)[/tex] hours.
2. Substitute the values into the formula:
- The formula becomes:
[tex]\[ P_t = 610 \cdot 2^{\frac{13}{6}} \][/tex]
3. Calculate the exponent:
- First, find [tex]\( \frac{13}{6} \)[/tex], which is approximately 2.1667.
4. Calculate [tex]\( 2^{\frac{13}{6}} \)[/tex]:
- This value is approximately 4.4901.
5. Multiply by the initial population:
- Now, multiply 610 by 4.4901:
[tex]\[ P_t = 610 \times 4.4901 \approx 2738.81 \][/tex]
6. Round to the nearest whole number:
- The population should be rounded to the nearest whole number, which is 2739.
So, the population of bacteria after 13 hours is approximately 2739.
[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, which is 610 bacteria.
- [tex]\( t \)[/tex] is the time in hours, which is 13 in this case.
- [tex]\( d \)[/tex] is the doubling time, which is 6 hours.
Let's go through the steps:
1. Identify the known values:
- Initial population, [tex]\( P_0 = 610 \)[/tex].
- Doubling time, [tex]\( d = 6 \)[/tex] hours.
- Time, [tex]\( t = 13 \)[/tex] hours.
2. Substitute the values into the formula:
- The formula becomes:
[tex]\[ P_t = 610 \cdot 2^{\frac{13}{6}} \][/tex]
3. Calculate the exponent:
- First, find [tex]\( \frac{13}{6} \)[/tex], which is approximately 2.1667.
4. Calculate [tex]\( 2^{\frac{13}{6}} \)[/tex]:
- This value is approximately 4.4901.
5. Multiply by the initial population:
- Now, multiply 610 by 4.4901:
[tex]\[ P_t = 610 \times 4.4901 \approx 2738.81 \][/tex]
6. Round to the nearest whole number:
- The population should be rounded to the nearest whole number, which is 2739.
So, the population of bacteria after 13 hours is approximately 2739.
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