A culture of bacteria has an initial population of 94,000 bacteria and doubles every 10 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?

Question

19-11-2023
Mathematics

High School

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A culture of bacteria has an initial population of 94,000 bacteria and doubles every 10 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 :

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noormalik3019 noormalik3019 · Answer 1 · 2023-07-21T22:34:05-04:00

Final Answer:

The population of bacteria in the culture after 13 hours is approximately 1,498,272 to the nearest whole number.

Explanation:

The formula for exponential growth of a population, [tex]$p_t = p_0 \times 2^{t/d}$, where $p_t$[/tex]is the population after [tex]$t$ hours, $p_0$[/tex]is the initial population, t is the time in hours, and d is the doubling time. Substituting the given values -[tex]$p_0 = 94000$, $t = 13$, and $d = 10$ -[/tex] into the formula yields \[tex](p_{13} = 94000 \times 2^{13/10}\).[/tex] Calculating this gives us approximately 1,498,272 bacteria after 13 hours of growth. This result is obtained by applying the formula to determine the exponential increase in the bacteria population after the specified time frame.

In this exponential growth scenario, the bacteria population undergoes doubling every 10 hours, and by utilizing the formula for exponential growth, we ascertain the population after 13 hours. The calculation involves the initial population of 94,000 bacteria and utilizes the exponential growth formula to determine the final count after the given time interval. This approach yields an approximate count of 1,498,272 bacteria in the culture after 13 hours.