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A culture of bacteria has an initial population of 230 bacteria and doubles every 9 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 :

Certainly! Let's solve the problem step by step using the formula provided for the bacteria population growth.

We are given:
- The initial population, [tex]\( P_0 = 230 \)[/tex] bacteria.
- The doubling time, [tex]\( d = 9 \)[/tex] hours.
- The time after which we want to find the population, [tex]\( t = 13 \)[/tex] hours.

The formula to calculate the population after [tex]\( t \)[/tex] hours is:
[tex]\[
P_t = P_0 \cdot 2^{\frac{t}{d}}
\][/tex]

Step 1: Identify the initial values.
- Initial population, [tex]\( P_0 = 230 \)[/tex]
- Time passed, [tex]\( t = 13 \)[/tex] hours
- Doubling time, [tex]\( d = 9 \)[/tex] hours

Step 2: Substitute the values into the formula.
[tex]\[
P_t = 230 \cdot 2^{\frac{13}{9}}
\][/tex]

Step 3: Calculate the exponent.
- First, calculate [tex]\( \frac{13}{9} \approx 1.4444 \)[/tex]

Step 4: Calculate the power of 2.
- [tex]\( 2^{1.4444} \approx 2.718 \)[/tex]

Step 5: Calculate the population after 13 hours.
- Multiply the initial population by the result from the previous step:
[tex]\[
P_t = 230 \cdot 2.718 \approx 625.963
\][/tex]

Step 6: Round the population to the nearest whole number.
- So, the population of bacteria after 13 hours is approximately 626.

Therefore, the population of the bacteria culture after 13 hours is 626.

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