Question

This exercise uses the population growth model. The count in a culture of bacteria was 400 after 2 hours and 25,600 after 6 hours. When will the number of bacteria be 50,000 ?

Answer 1

To solve for the time when the number of bacteria will be 50,000, we can use the formula for exponential growth. By plugging in the given values and solving for time, we find that it will take approximately 7.124 hours.

Step-by-step explanation:

To solve this problem, we need to use the formula for exponential growth: N = N0 * e^(rt), where N is the final population size, N0 is the initial population size, e is the base of natural logarithms, r is the growth rate, and t is the time.

Let's plug in the given values:

N0 = 400 (after 2 hours)

N = 25600 (after 6 hours)

We can solve for the growth rate (r) by dividing the two equations:

r = ln(N/N0) / t

r = ln(25600/400) / 6

r ≈ 0.219

Now we can plug in the known values and solve for the time (t) when the number of bacteria will be 50000:

N = N0 * e^(rt)

50000 = 400 * e^(0.219t)

e^(0.219t) = 50000 / 400

e^(0.219t) ≈ 125

0.219t ≈ ln(125)

t ≈ ln(125) / 0.219

Using a calculator, we find that t ≈ 7.124 hours.