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A culture of bacteria has an initial population of 94,000 bacteria and doubles every 10 hours. 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, and [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 find the population of bacteria after 13 hours, we can use the given formula for population growth:

[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 (94,000 bacteria),
- [tex]\( t \)[/tex] is the time in hours (13 hours),
- [tex]\( d \)[/tex] is the doubling time (10 hours).

Here's how we can calculate it step-by-step:

1. Identify the initial values:
- Initial population, [tex]\( P_0 = 94,000 \)[/tex]
- Time, [tex]\( t = 13 \)[/tex] hours
- Doubling time, [tex]\( d = 10 \)[/tex] hours

2. Plug the values into the formula:

[tex]\[
P_t = 94,000 \cdot 2^{\frac{13}{10}}
\][/tex]

3. Calculate the exponent:

[tex]\[
\frac{13}{10} = 1.3
\][/tex]

4. Calculate the power of 2:

[tex]\[
2^{1.3} \approx 2.462
\][/tex]

5. Multiply by the initial population:

[tex]\[
P_t = 94,000 \cdot 2.462
\][/tex]

[tex]\[
P_t \approx 231,455
\][/tex]

6. Round the result to the nearest whole number.

Therefore, the population of bacteria after 13 hours is approximately 231,455 bacteria.

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