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Answer :
Sure! Let's find the population of bacteria in the culture after 13 hours using the given formula.
Step 1: Understand the Formula
The formula we will use is:
[tex]\[ P_t = P_0 \cdot 2^{\frac{t}{d}} \][/tex]
- [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.
Step 2: Identify the Given Values
We know:
- The initial population, [tex]\( P_0 = 610 \)[/tex] bacteria.
- The time, [tex]\( t = 13 \)[/tex] hours.
- The doubling time, [tex]\( d = 6 \)[/tex] hours.
Step 3: Plug the Values into the Formula
Substitute the known values into the formula:
[tex]\[ P_t = 610 \cdot 2^{\frac{13}{6}} \][/tex]
Step 4: Calculate the Exponent
First, solve the fraction within the exponent:
[tex]\[ \frac{13}{6} \approx 2.1667 \][/tex] (rounded to four decimal places)
Step 5: Calculate the Power of 2
Next, calculate [tex]\( 2^{2.1667} \)[/tex]. This is approximately:
[tex]\[ 2^{2.1667} \approx 4.4902 \][/tex]
Step 6: Calculate the Population
Now, multiply this result by the initial population:
[tex]\[ P_t = 610 \times 4.4902 \][/tex]
[tex]\[ P_t \approx 2740 \][/tex]
Conclusion
Therefore, the population of bacteria after 13 hours is approximately 2740.
Step 1: Understand the Formula
The formula we will use is:
[tex]\[ P_t = P_0 \cdot 2^{\frac{t}{d}} \][/tex]
- [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.
Step 2: Identify the Given Values
We know:
- The initial population, [tex]\( P_0 = 610 \)[/tex] bacteria.
- The time, [tex]\( t = 13 \)[/tex] hours.
- The doubling time, [tex]\( d = 6 \)[/tex] hours.
Step 3: Plug the Values into the Formula
Substitute the known values into the formula:
[tex]\[ P_t = 610 \cdot 2^{\frac{13}{6}} \][/tex]
Step 4: Calculate the Exponent
First, solve the fraction within the exponent:
[tex]\[ \frac{13}{6} \approx 2.1667 \][/tex] (rounded to four decimal places)
Step 5: Calculate the Power of 2
Next, calculate [tex]\( 2^{2.1667} \)[/tex]. This is approximately:
[tex]\[ 2^{2.1667} \approx 4.4902 \][/tex]
Step 6: Calculate the Population
Now, multiply this result by the initial population:
[tex]\[ P_t = 610 \times 4.4902 \][/tex]
[tex]\[ P_t \approx 2740 \][/tex]
Conclusion
Therefore, the population of bacteria after 13 hours is approximately 2740.
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