High School

A culture of bacteria has an initial population of 400 bacteria 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?

A) 1,600
B) 3,200
C) 6,400
D) 12,800

Answer :

Rounded to the nearest whole number, the population of bacteria after 19 hours is approximately 28915.

To find the population of bacteria after 19 hours, we can use the formula for exponential growth:

[tex]\[ P(t) = P_0 \times 2^{(t / T)} \][/tex]

Where:

P(t) = population at time t

[tex]P_0[/tex] = initial population

t = time elapsed

T = doubling time

Given:

P_0 = 4600 (initial population)

T = 10 hours (doubling time)

t = 19 hours (time elapsed)

Plugging these values into the formula:

[tex]\[ P(19) = 4600 \times 2^{(19 / 10)} \]\[ P(19) = 4600 \times 2^{1.9} \]\[ P(19) = 4600 \times 6.279 \]\[ P(19) \approx 28915.4 \][/tex]

Rounded to the nearest whole number, the population of bacteria after 19 hours is approximately 28915.

Question

A culture of bacteria has an initial population of 4600 bacteria and doubles every 10 hours. Using the formula what is the population of bacteria in the culture after 19 hours, to the nearest whole number?

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