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Answer :
To determine the population of bacteria in a culture after a given period of time, we will use the formula for exponential growth:
[tex]\[ P_t = P_0 \cdot 2^{t/d} \][/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 in hours.
For this specific problem, the initial population ([tex]\( P_0 \)[/tex]) is 65,000 bacteria, the time ([tex]\( t \)[/tex]) is 13 hours, and the doubling time ([tex]\( d \)[/tex]) is 2 hours. We'll substitute these values into the formula to find the population after 13 hours.
Step-by-step solution:
1. Identify the given values:
- [tex]\( P_0 = 65000 \)[/tex]
- [tex]\( t = 13 \)[/tex]
- [tex]\( d = 2 \)[/tex]
2. Plug these values into the formula [tex]\( P_t = P_0 \cdot 2^{t/d} \)[/tex]:
[tex]\[ P_t = 65000 \cdot 2^{13/2} \][/tex]
3. Calculate the exponent:
[tex]\[ \frac{13}{2} = 6.5 \][/tex]
4. Calculate [tex]\( 2^{6.5} \)[/tex]:
[tex]\[ 2^{6.5} \approx 90.50967 \][/tex]
5. Multiply the initial population by [tex]\( 2^{6.5} \)[/tex]:
[tex]\[ P_t = 65000 \cdot 90.50967 \][/tex]
[tex]\[ P_t \approx 5883128.419472076 \][/tex]
6. Finally, round this population to the nearest whole number:
[tex]\[ P_t \approx 5883128 \][/tex]
Therefore, the population of bacteria in the culture after 13 hours, rounded to the nearest whole number, is approximately 5,883,128 bacteria.
[tex]\[ P_t = P_0 \cdot 2^{t/d} \][/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 in hours.
For this specific problem, the initial population ([tex]\( P_0 \)[/tex]) is 65,000 bacteria, the time ([tex]\( t \)[/tex]) is 13 hours, and the doubling time ([tex]\( d \)[/tex]) is 2 hours. We'll substitute these values into the formula to find the population after 13 hours.
Step-by-step solution:
1. Identify the given values:
- [tex]\( P_0 = 65000 \)[/tex]
- [tex]\( t = 13 \)[/tex]
- [tex]\( d = 2 \)[/tex]
2. Plug these values into the formula [tex]\( P_t = P_0 \cdot 2^{t/d} \)[/tex]:
[tex]\[ P_t = 65000 \cdot 2^{13/2} \][/tex]
3. Calculate the exponent:
[tex]\[ \frac{13}{2} = 6.5 \][/tex]
4. Calculate [tex]\( 2^{6.5} \)[/tex]:
[tex]\[ 2^{6.5} \approx 90.50967 \][/tex]
5. Multiply the initial population by [tex]\( 2^{6.5} \)[/tex]:
[tex]\[ P_t = 65000 \cdot 90.50967 \][/tex]
[tex]\[ P_t \approx 5883128.419472076 \][/tex]
6. Finally, round this population to the nearest whole number:
[tex]\[ P_t \approx 5883128 \][/tex]
Therefore, the population of bacteria in the culture after 13 hours, rounded to the nearest whole number, is approximately 5,883,128 bacteria.
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