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Answer :
To solve the problem of finding 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,
- [tex]\( t \)[/tex] is the time in hours, and
- [tex]\( d \)[/tex] is the doubling time in hours.
Let's break down the solution step-by-step:
1. Identify the given 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:
[tex]\[ P_t = 610 \cdot 2^{\frac{13}{6}} \][/tex]
3. Calculate the exponent:
- First, divide the time by the doubling time:
[tex]\[ \frac{13}{6} \approx 2.1667 \][/tex]
4. Evaluate the power of 2:
- Calculate [tex]\( 2^{2.1667} \)[/tex].
5. Calculate the population:
- Multiply the initial population by the result from step 4:
[tex]\[ P_t \approx 610 \cdot 4.492 \][/tex]
- Here we get the approximate population value: [tex]\[ 2738.807 \][/tex]
6. Round to the nearest whole number:
- Round 2738.807 to the nearest whole number: [tex]\[ P_t \approx 2739 \][/tex]
This means 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,
- [tex]\( t \)[/tex] is the time in hours, and
- [tex]\( d \)[/tex] is the doubling time in hours.
Let's break down the solution step-by-step:
1. Identify the given 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:
[tex]\[ P_t = 610 \cdot 2^{\frac{13}{6}} \][/tex]
3. Calculate the exponent:
- First, divide the time by the doubling time:
[tex]\[ \frac{13}{6} \approx 2.1667 \][/tex]
4. Evaluate the power of 2:
- Calculate [tex]\( 2^{2.1667} \)[/tex].
5. Calculate the population:
- Multiply the initial population by the result from step 4:
[tex]\[ P_t \approx 610 \cdot 4.492 \][/tex]
- Here we get the approximate population value: [tex]\[ 2738.807 \][/tex]
6. Round to the nearest whole number:
- Round 2738.807 to the nearest whole number: [tex]\[ P_t \approx 2739 \][/tex]
This means the population of bacteria after 13 hours is approximately 2739.
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