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Answer :
To find the population of a culture of bacteria after 14 hours, if the initial population is 790 bacteria and the population doubles every 5 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 time [tex]\( t \)[/tex].
- [tex]\( P_0 \)[/tex] is the initial population.
- [tex]\( t \)[/tex] is the time in hours.
- [tex]\( d \)[/tex] is the doubling time.
Given values:
- [tex]\( P_0 = 790 \)[/tex]
- [tex]\( t = 14 \)[/tex] hours
- [tex]\( d = 5 \)[/tex] hours
Now, let's substitute these values into the formula:
[tex]\[ P_t = 790 \cdot 2^{\frac{14}{5}} \][/tex]
First, calculate [tex]\(\frac{14}{5}\)[/tex]:
[tex]\[\frac{14}{5} = 2.8\][/tex]
Now, calculate [tex]\(2^{2.8}\)[/tex]:
Calculating [tex]\(2^{2.8}\)[/tex], we find that the result is approximately 6.962.
Now, substitute back into the expression:
[tex]\[ P_t = 790 \cdot 6.962 \][/tex]
Finally, calculate this multiplication:
[tex]\[ P_t \approx 5501.88 \][/tex]
When rounded to the nearest whole number, the population is approximately 5502.
Therefore, the population of bacteria in the culture after 14 hours is approximately 5502.
[tex]\[ P_t = P_0 \cdot 2^{\frac{t}{d}} \][/tex]
Where:
- [tex]\( P_t \)[/tex] is the population after time [tex]\( t \)[/tex].
- [tex]\( P_0 \)[/tex] is the initial population.
- [tex]\( t \)[/tex] is the time in hours.
- [tex]\( d \)[/tex] is the doubling time.
Given values:
- [tex]\( P_0 = 790 \)[/tex]
- [tex]\( t = 14 \)[/tex] hours
- [tex]\( d = 5 \)[/tex] hours
Now, let's substitute these values into the formula:
[tex]\[ P_t = 790 \cdot 2^{\frac{14}{5}} \][/tex]
First, calculate [tex]\(\frac{14}{5}\)[/tex]:
[tex]\[\frac{14}{5} = 2.8\][/tex]
Now, calculate [tex]\(2^{2.8}\)[/tex]:
Calculating [tex]\(2^{2.8}\)[/tex], we find that the result is approximately 6.962.
Now, substitute back into the expression:
[tex]\[ P_t = 790 \cdot 6.962 \][/tex]
Finally, calculate this multiplication:
[tex]\[ P_t \approx 5501.88 \][/tex]
When rounded to the nearest whole number, the population is approximately 5502.
Therefore, the population of bacteria in the culture after 14 hours is approximately 5502.
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