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Answer :
Certainly! Let's solve the problem step by step using the formula provided for the bacteria population growth.
We are given:
- The initial population, [tex]\( P_0 = 230 \)[/tex] bacteria.
- The doubling time, [tex]\( d = 9 \)[/tex] hours.
- The time after which we want to find the population, [tex]\( t = 13 \)[/tex] hours.
The formula to calculate the population after [tex]\( t \)[/tex] hours is:
[tex]\[
P_t = P_0 \cdot 2^{\frac{t}{d}}
\][/tex]
Step 1: Identify the initial values.
- Initial population, [tex]\( P_0 = 230 \)[/tex]
- Time passed, [tex]\( t = 13 \)[/tex] hours
- Doubling time, [tex]\( d = 9 \)[/tex] hours
Step 2: Substitute the values into the formula.
[tex]\[
P_t = 230 \cdot 2^{\frac{13}{9}}
\][/tex]
Step 3: Calculate the exponent.
- First, calculate [tex]\( \frac{13}{9} \approx 1.4444 \)[/tex]
Step 4: Calculate the power of 2.
- [tex]\( 2^{1.4444} \approx 2.718 \)[/tex]
Step 5: Calculate the population after 13 hours.
- Multiply the initial population by the result from the previous step:
[tex]\[
P_t = 230 \cdot 2.718 \approx 625.963
\][/tex]
Step 6: Round the population to the nearest whole number.
- So, the population of bacteria after 13 hours is approximately 626.
Therefore, the population of the bacteria culture after 13 hours is 626.
We are given:
- The initial population, [tex]\( P_0 = 230 \)[/tex] bacteria.
- The doubling time, [tex]\( d = 9 \)[/tex] hours.
- The time after which we want to find the population, [tex]\( t = 13 \)[/tex] hours.
The formula to calculate the population after [tex]\( t \)[/tex] hours is:
[tex]\[
P_t = P_0 \cdot 2^{\frac{t}{d}}
\][/tex]
Step 1: Identify the initial values.
- Initial population, [tex]\( P_0 = 230 \)[/tex]
- Time passed, [tex]\( t = 13 \)[/tex] hours
- Doubling time, [tex]\( d = 9 \)[/tex] hours
Step 2: Substitute the values into the formula.
[tex]\[
P_t = 230 \cdot 2^{\frac{13}{9}}
\][/tex]
Step 3: Calculate the exponent.
- First, calculate [tex]\( \frac{13}{9} \approx 1.4444 \)[/tex]
Step 4: Calculate the power of 2.
- [tex]\( 2^{1.4444} \approx 2.718 \)[/tex]
Step 5: Calculate the population after 13 hours.
- Multiply the initial population by the result from the previous step:
[tex]\[
P_t = 230 \cdot 2.718 \approx 625.963
\][/tex]
Step 6: Round the population to the nearest whole number.
- So, the population of bacteria after 13 hours is approximately 626.
Therefore, the population of the bacteria culture after 13 hours is 626.
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