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Answer :
To solve the problem of finding the population of bacteria after 13 hours, we use the provided 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,
- [tex]\( d \)[/tex] is the doubling time.
Given:
- The initial population [tex]\( P_0 \)[/tex] is 94,000 bacteria.
- The doubling time [tex]\( d \)[/tex] is 10 hours.
- We want to find the population after [tex]\( t = 13 \)[/tex] hours.
Here's how you calculate it:
1. Substitute the values into the formula:
[tex]\[ P_t = 94000 \cdot 2^{\frac{13}{10}} \][/tex]
2. Calculate the exponent:
[tex]\[ \frac{13}{10} = 1.3 \][/tex]
3. Evaluate [tex]\( 2^{1.3} \)[/tex]. Approximating, this gives:
[tex]\[ 2^{1.3} \approx 2.46 \][/tex]
4. Multiply the initial population by this result:
[tex]\[ P_t = 94000 \times 2.46 \][/tex]
5. This calculation results in approximately 231,455.15.
6. Finally, round to the nearest whole number:
[tex]\[ P_t \approx 231,455 \][/tex]
Therefore, the population of bacteria after 13 hours is approximately 231,455.
[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,
- [tex]\( d \)[/tex] is the doubling time.
Given:
- The initial population [tex]\( P_0 \)[/tex] is 94,000 bacteria.
- The doubling time [tex]\( d \)[/tex] is 10 hours.
- We want to find the population after [tex]\( t = 13 \)[/tex] hours.
Here's how you calculate it:
1. Substitute the values into the formula:
[tex]\[ P_t = 94000 \cdot 2^{\frac{13}{10}} \][/tex]
2. Calculate the exponent:
[tex]\[ \frac{13}{10} = 1.3 \][/tex]
3. Evaluate [tex]\( 2^{1.3} \)[/tex]. Approximating, this gives:
[tex]\[ 2^{1.3} \approx 2.46 \][/tex]
4. Multiply the initial population by this result:
[tex]\[ P_t = 94000 \times 2.46 \][/tex]
5. This calculation results in approximately 231,455.15.
6. Finally, round to the nearest whole number:
[tex]\[ P_t \approx 231,455 \][/tex]
Therefore, the population of bacteria after 13 hours is approximately 231,455.
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