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A culture of bacteria has an initial population of 65,000 bacteria and doubles every 2 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,
- [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 given the initial population and doubling time, you can follow these steps using the provided formula:

1. Determine the Initial Population and Doubling Time:
- Initial population [tex]\( P_0 = 65,000 \)[/tex] bacteria.
- Doubling time [tex]\( d = 2 \)[/tex] hours.

2. Identify the Time Duration:
- The time after which we need to find the population is [tex]\( t = 13 \)[/tex] hours.

3. Use the Population Growth Formula:
- The formula to calculate the population at time [tex]\( t \)[/tex] is:
[tex]\[
P_t = P_0 \cdot 2^{\frac{t}{d}}
\][/tex]

4. Plug in the Values into the Formula:
- Substitute the given values into the formula:
[tex]\[
P_t = 65,000 \cdot 2^{\frac{13}{2}}
\][/tex]

5. Calculate the Exponent:
- The exponent is [tex]\( \frac{13}{2} = 6.5 \)[/tex].

6. Calculate the Population:
- Compute [tex]\( 2^{6.5} \)[/tex] and then multiply by 65,000.

7. Result:
- The population of bacteria after 13 hours is approximately 5,883,128 when rounded to the nearest whole number.

Therefore, the population of bacteria in the culture after 13 hours is approximately 5,883,128 bacteria.

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