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Answer :
To calculate the population of bacteria after 13 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]\( P_0 = 230 \)[/tex] is the initial population.
- [tex]\( t = 13 \)[/tex] is the time in hours.
- [tex]\( d = 9 \)[/tex] is the doubling time in hours.
Let's break it down step-by-step:
1. Substitute the values into the formula:
[tex]\[
P_t = 230 \cdot 2^{\frac{13}{9}}
\][/tex]
2. Calculate the exponent:
First, calculate the exponent [tex]\( \frac{13}{9} \)[/tex]. When you do this calculation, you find that [tex]\( \frac{13}{9} \approx 1.4444 \)[/tex].
3. Calculate the power of 2:
Now, calculate [tex]\( 2^{1.4444} \)[/tex]. This results in approximately 2.743.
4. Multiply by the initial population:
Multiply the initial population [tex]\( 230 \)[/tex] by approximately [tex]\( 2.743 \)[/tex]:
[tex]\[
230 \cdot 2.743 \approx 625.963
\][/tex]
5. Round to the nearest whole number:
Finally, round 625.963 to the nearest whole number, which gives you:
[tex]\[
P_t \approx 626
\][/tex]
Therefore, the population of bacteria in the culture after 13 hours is approximately 626.
[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]\( P_0 = 230 \)[/tex] is the initial population.
- [tex]\( t = 13 \)[/tex] is the time in hours.
- [tex]\( d = 9 \)[/tex] is the doubling time in hours.
Let's break it down step-by-step:
1. Substitute the values into the formula:
[tex]\[
P_t = 230 \cdot 2^{\frac{13}{9}}
\][/tex]
2. Calculate the exponent:
First, calculate the exponent [tex]\( \frac{13}{9} \)[/tex]. When you do this calculation, you find that [tex]\( \frac{13}{9} \approx 1.4444 \)[/tex].
3. Calculate the power of 2:
Now, calculate [tex]\( 2^{1.4444} \)[/tex]. This results in approximately 2.743.
4. Multiply by the initial population:
Multiply the initial population [tex]\( 230 \)[/tex] by approximately [tex]\( 2.743 \)[/tex]:
[tex]\[
230 \cdot 2.743 \approx 625.963
\][/tex]
5. Round to the nearest whole number:
Finally, round 625.963 to the nearest whole number, which gives you:
[tex]\[
P_t \approx 626
\][/tex]
Therefore, the population of bacteria in the culture after 13 hours is approximately 626.
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