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A culture of bacteria has an initial population of 610 bacteria and doubles every 6 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, we can use 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, which is 610 bacteria.
- [tex]\( t \)[/tex] is the time in hours, which is 13 in this case.
- [tex]\( d \)[/tex] is the doubling time, which is 6 hours.

Let's go through the steps:

1. Identify the known values:
- Initial population, [tex]\( P_0 = 610 \)[/tex].
- Doubling time, [tex]\( d = 6 \)[/tex] hours.
- Time, [tex]\( t = 13 \)[/tex] hours.

2. Substitute the values into the formula:
- The formula becomes:

[tex]\[ P_t = 610 \cdot 2^{\frac{13}{6}} \][/tex]

3. Calculate the exponent:
- First, find [tex]\( \frac{13}{6} \)[/tex], which is approximately 2.1667.

4. Calculate [tex]\( 2^{\frac{13}{6}} \)[/tex]:
- This value is approximately 4.4901.

5. Multiply by the initial population:
- Now, multiply 610 by 4.4901:

[tex]\[ P_t = 610 \times 4.4901 \approx 2738.81 \][/tex]

6. Round to the nearest whole number:
- The population should be rounded to the nearest whole number, which is 2739.

So, the population of bacteria after 13 hours is approximately 2739.

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