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A culture of bacteria has an initial population of 430. Using 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,

What is the population of bacteria in the culture after 13 hours, to the nearest whole number?

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.

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