Question

The population of bacteria in a jar grows at a rate proportional to the number of bacteria present at time t. Initially the jar has 20 bacteria. After 8 hours, it is observed that 50 bacteria are present. How many bacteria will there be after 72 hours? You need to set up the differential equation and give an explicit solution.

Answer 1

Answer: Therefore, there will be 15,625 bacteria after 72 hours.

Step-by-step explanation:

Let N(t) be the number of bacteria at time t. The rate of change of bacteria is proportional to the number of bacteria present, so we have:

dN/dt = kN

where k is the constant of proportionality.

Using the initial condition N(0) = 20, we can solve for k:

N(0) = 20

N(t) = N(0)e^(kt)

50 = 20e^(8k)

ln(50/20) = 8k

k = ln(5/2)/8

Now, we can find N(72):

N(72) = 20e^(72k)

N(72) = 20e^(9ln(5/2))

N(72) = 20(5/2)^9

N(72) = 15625

Therefore, there will be 15,625 bacteria after 72 hours.