High School

A culture of bacteria has an initial population of 43,000 bacteria and doubles every 5 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 formula:

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

where:
- [tex]\( P_0 \)[/tex] is the initial population, which is 43,000 bacteria.
- [tex]\( t \)[/tex] is the time in hours, which is 13 hours in this case.
- [tex]\( d \)[/tex] is the doubling time, which is 5 hours.

Let's break it down step-by-step:

1. Identify the initial values:
- Initial population [tex]\( P_0 = 43,000 \)[/tex]
- Time [tex]\( t = 13 \)[/tex] hours
- Doubling time [tex]\( d = 5 \)[/tex] hours

2. Plug the values into the formula:

[tex]\[ P_t = 43,000 \cdot 2^{\frac{13}{5}} \][/tex]

3. Calculate the exponent:

[tex]\[ \frac{13}{5} = 2.6 \][/tex]

4. Calculate [tex]\( 2^{2.6} \)[/tex]:

- [tex]\( 2^{2.6} \)[/tex] is approximately equal to 6.035

5. Now calculate [tex]\( P_t \)[/tex]:

[tex]\[ P_t = 43,000 \cdot 6.035 \][/tex]
[tex]\[ P_t \approx 260,703.25 \][/tex]

6. Round to the nearest whole number:

The population, when rounded, is approximately 260,703.

Therefore, the population of bacteria after 13 hours is approximately 260,703 bacteria.

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