During a research experiment it was found that the number of

During a research experiment, it was found that the number of bacteria in a culture grew at a rate proportional to its size. At 9:00 AM there were 3,000 bacteria present in the culture. At noon, the number of bacteria grew to 3,500. How many bacteria will there be at midnight?

Solution

dB/dt = k*B

dB/kB = dt

(1/k)*ln B = t + ln C

B = C*e^(k*t)

At t = 0

B = 3000

3000 = C*e^0

C = 3000

at t = 3

B = 3500

3500 = 3000*e^3k

e^3k = 7/6

k = (1/3)*ln (7/6)

k = 0.0514

So, when t = 15

B = 3000*e^0.0514*15

B = 6485.78 baxteria at midnight.