High School

A culture of bacteria has an initial population of 980 bacteria and doubles every 9 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, and
- \( d \) is the doubling time,

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

Answer :

Answer: pt=2667.1484≈2667

The population of bacteria in the culture after 13 hours is approximately 2704 bacteria, rounded to the nearest whole number.

To calculate the population of bacteria in the culture after 13 hours, using the provided formula [tex]P_t = P_0 \cdot 2^{\frac{t}{d}}[/tex], we can plug in the values: [tex]P_0[/tex] (initial population) = 980, t (time in hours after start) = 13, and d (doubling time in hours) = 9. The calculation will look like this:

[tex]P_t = 980 \cdot 2^{\frac{13}{9}}[/tex]

This formula represents exponential growth, where the population doubles after every set period of time, in this case, every 9 hours. The solution is as follows:

  • Calculate the exponent: 13/9 ≈ 1.4444.
  • Calculate 2 to the power of 1.4444.
  • Multiply the initial population by this value: 980 * [tex]2^{1.4444}[/tex].

Using a calculator, we find:

  • [tex]2^{1.4444}[/tex] ≈ 2.7595.
  • Therefore, [tex]P_t[/tex] ≈ 980 * 2.7595 ≈ 2704.31.

To the nearest whole number, the population of bacteria after 13 hours would be 2704.

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