High School

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 :

Sure! Let's find the population of bacteria in the culture after 13 hours using the given formula.

Step 1: Understand the Formula

The formula we will use is:

[tex]\[ P_t = P_0 \cdot 2^{\frac{t}{d}} \][/tex]

- [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.

Step 2: Identify the Given Values

We know:
- The initial population, [tex]\( P_0 = 610 \)[/tex] bacteria.
- The time, [tex]\( t = 13 \)[/tex] hours.
- The doubling time, [tex]\( d = 6 \)[/tex] hours.

Step 3: Plug the Values into the Formula

Substitute the known values into the formula:

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

Step 4: Calculate the Exponent

First, solve the fraction within the exponent:

[tex]\[ \frac{13}{6} \approx 2.1667 \][/tex] (rounded to four decimal places)

Step 5: Calculate the Power of 2

Next, calculate [tex]\( 2^{2.1667} \)[/tex]. This is approximately:

[tex]\[ 2^{2.1667} \approx 4.4902 \][/tex]

Step 6: Calculate the Population

Now, multiply this result by the initial population:

[tex]\[ P_t = 610 \times 4.4902 \][/tex]

[tex]\[ P_t \approx 2740 \][/tex]

Conclusion

Therefore, the population of bacteria after 13 hours is approximately 2740.

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