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A culture of bacteria has an initial population of 65,000 and doubles every 2 hours. Using the formula

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

where:
- \( P_t \) is the population after \( t \) hours,
- \( P_0 \) is the initial population,
- \( t \) is the time in hours,
- \( d \) is the doubling time,

what is the population of bacteria in the culture after 13 hours, to the nearest whole number?

Answer :

To solve this problem, you will use the formula for exponential growth:

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

Where:

  • [tex]P_t[/tex] is the population at time [tex]t[/tex] hours.
  • [tex]P_0[/tex] is the initial population, which is 65,000 bacteria in this problem.
  • [tex]t[/tex] is the time in hours, which is 13 hours in this problem.
  • [tex]d[/tex] is the doubling time, which is 2 hours for the bacteria.

Follow these steps to find the bacterial population after 13 hours:

  1. Identify the Known Values:

    • Initial population, [tex]P_0 = 65,000[/tex]
    • Doubling time, [tex]d = 2[/tex] hours
    • Time elapsed, [tex]t = 13[/tex] hours

  2. Substitute Known Values into the Formula:
    [tex]P_{13} = 65,000 \cdot 2^{\frac{13}{2}}[/tex]

  3. Calculate the Exponent:

    • [tex]\frac{13}{2} = 6.5[/tex]

  4. Compute the Power of 2:

    • [tex]2^{6.5} \approx 90.51[/tex]

  5. Multiply by the Initial Population:

    • [tex]P_{13} = 65,000 \times 90.51[/tex]
    • ( P_{13} \approx 5,883,150

Therefore, the population of bacteria after 13 hours is approximately 5,883,150 when rounded to the nearest whole number.

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