High School

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, and \( d \) is the doubling time, what is the population of bacteria in the culture after 13 hours, to the nearest whole number?

Answer :

Final answer:

To calculate the bacterial population after 13 hours, use the exponential growth formula and plug in the values: initial population (65,000), time (13), and doubling interval (2 hours). After calculations, the population is approximately 5,883,150 bacteria.

Explanation:

The student's question regards the population growth of bacteria, which is a concept in mathematics that applies to biology. Using the provided formula for exponential growth pt = p0 · 2t/d, where p0 is the initial population, t is the time in hours, d is the doubling time, and pt is the population after t hours, we can calculate the population of bacteria after 13 hours.

The initial population (p0) is 65,000 bacteria, and the doubling time (d) is 2 hours. To find the population after 13 hours (t), we use the formula with these values:

pt = 65,000 · 213/2

First, divide 13 hours by the doubling time of 2 hours:

13/2 = 6.5

The population doubles 6.5 times in 13 hours. We calculate the doubling factor:

26.5 ≈ 90.510

Next, multiply the initial population by this factor:

pt = 65,000 · 90.510 ≈ 5,883,150

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

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