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Answer :
To find the population of bacteria after 13 hours, given that the initial population is 43,000 and it 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 [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.
Now, let's plug in the values:
- [tex]\( P_0 = 43,000 \)[/tex]
- [tex]\( t = 13 \)[/tex] hours,
- [tex]\( d = 5 \)[/tex] hours.
Substitute these values into the formula:
[tex]\[ P_t = 43,000 \cdot 2^{\frac{13}{5}} \][/tex]
First, calculate the exponent [tex]\(\frac{13}{5}\)[/tex]:
[tex]\(\frac{13}{5} = 2.6\)[/tex]
Now, calculate [tex]\(2^{2.6}\)[/tex]:
[tex]\(2^{2.6} \approx 6.305\)[/tex]
Next, multiply the initial population by this result:
[tex]\[ P_t \approx 43,000 \cdot 6.305 \][/tex]
[tex]\[ P_t \approx 271,115.0\][/tex]
Round this to the nearest whole number:
So, the population of bacteria in the culture after 13 hours is approximately 260,703 bacteria.
[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 in hours.
Now, let's plug in the values:
- [tex]\( P_0 = 43,000 \)[/tex]
- [tex]\( t = 13 \)[/tex] hours,
- [tex]\( d = 5 \)[/tex] hours.
Substitute these values into the formula:
[tex]\[ P_t = 43,000 \cdot 2^{\frac{13}{5}} \][/tex]
First, calculate the exponent [tex]\(\frac{13}{5}\)[/tex]:
[tex]\(\frac{13}{5} = 2.6\)[/tex]
Now, calculate [tex]\(2^{2.6}\)[/tex]:
[tex]\(2^{2.6} \approx 6.305\)[/tex]
Next, multiply the initial population by this result:
[tex]\[ P_t \approx 43,000 \cdot 6.305 \][/tex]
[tex]\[ P_t \approx 271,115.0\][/tex]
Round this to the nearest whole number:
So, the population of bacteria in the culture after 13 hours is approximately 260,703 bacteria.
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