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Answer :
To solve the problem of finding the population of bacteria in the culture after 13 hours, we can use the provided formula:
[tex]\[ P_t = P_0 \cdot 2^{\frac{t}{d}} \][/tex]
Here's a step-by-step breakdown:
1. Identify the Given Information:
- The initial population ([tex]\(P_0\)[/tex]) is 430.
- The time ([tex]\(t\)[/tex]) for which we need to calculate the population is 13 hours.
- The doubling time ([tex]\(d\)[/tex]) is assumed to be 2 hours, which is a typical doubling time for bacteria.
2. Apply the Formula:
- Substitute the given values into the formula:
[tex]\[ P_t = 430 \cdot 2^{\frac{13}{2}} \][/tex]
3. Calculate the Exponent:
- [tex]\( \frac{13}{2} = 6.5 \)[/tex]
4. Calculate [tex]\(2^{6.5}\)[/tex]:
- This step involves calculating the power of 2 raised to 6.5.
5. Calculate [tex]\(P_t\)[/tex]:
- Multiply the initial population by the result of the exponentiation:
[tex]\[ P_t = 430 \cdot 2^{6.5} \][/tex]
6. Round the Result:
- After calculating the value, round the population to the nearest whole number.
7. Final Answer:
- The population of bacteria after 13 hours is approximately 38,919.
There you have it! The approximate population of the bacteria after 13 hours is 38,919.
[tex]\[ P_t = P_0 \cdot 2^{\frac{t}{d}} \][/tex]
Here's a step-by-step breakdown:
1. Identify the Given Information:
- The initial population ([tex]\(P_0\)[/tex]) is 430.
- The time ([tex]\(t\)[/tex]) for which we need to calculate the population is 13 hours.
- The doubling time ([tex]\(d\)[/tex]) is assumed to be 2 hours, which is a typical doubling time for bacteria.
2. Apply the Formula:
- Substitute the given values into the formula:
[tex]\[ P_t = 430 \cdot 2^{\frac{13}{2}} \][/tex]
3. Calculate the Exponent:
- [tex]\( \frac{13}{2} = 6.5 \)[/tex]
4. Calculate [tex]\(2^{6.5}\)[/tex]:
- This step involves calculating the power of 2 raised to 6.5.
5. Calculate [tex]\(P_t\)[/tex]:
- Multiply the initial population by the result of the exponentiation:
[tex]\[ P_t = 430 \cdot 2^{6.5} \][/tex]
6. Round the Result:
- After calculating the value, round the population to the nearest whole number.
7. Final Answer:
- The population of bacteria after 13 hours is approximately 38,919.
There you have it! The approximate population of the bacteria after 13 hours is 38,919.
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