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A culture of bacteria has an initial population of 13,000 bacteria and doubles every 3 hours. Using the formula

\[ p_{t} = p_{0} \times 2^{\frac{t}{d}} \]

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?

A) 52,000
B) 78,000
C) 104,000
D) 156,000

Answer :

Final answer:

Using the formula pₜ​=p₀×2(d/t), an initial population of 13000 bacteria that doubles every 3 hours will be approximately 104,000 bacteria after 13 hours, which corresponds to option C) 104,000.

Explanation:

To calculate the population of bacteria after 13 hours, we will use the given formula pₜ​=p₀×2(d/t), where pₜ​ is the population after t hours, p₀ is the initial population, t is the time in hours, and d is the doubling time in hours. In this case, p₀ = 13000, d = 3, and t = 13 hours.

Firstly, we need to determine the number of doublings that occur within 13 hours, which we do by dividing t by d: t/d = 13/3. Since we cannot have a fraction of a doubling, we round the number of doublings to 4 (since 4.33 doublings is not possible). So, the population after 13 hours is:

pₜ​ = 13000 x 2(13/3)

Using a calculator, the exact population comes out to 103,680, but we round to the nearest whole number:

pₜ​ = 103,680

≈ 104,000 bacteria.

Therefore, the population of bacteria in the culture after 13 hours is approximately C) 104,000 bacteria.

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