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 3000 bacteria present in the culture. At noon, the number of bacteria grew to 3800. How many bacteria will there be at midnight?

DB/dt = k*B

dB / (kB) = dt

Integrating both sides:

1/k*ln|B| = t + C

B = C*e^ (kt)

At time t = 0, B = 4000 and at time t = 3 (3 hours past 9 AM), B = 4500

4000 = C

4500 = 4000*e^ (3k) = > k = 1/3*ln (9/8)

At time t = 15 (Midnight), B = 4000*e^ (1/3*ln (9/8) * 15) = 4000 * (9/8) ^5 = 7208 bacteria