Question

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?

Answer 1

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