Question

Consider a population of bacteria that grows according to the initial value problem dP/dt=P/10, P(0)=300. Find the population size after 40 hours

Answer 1

dP / dt = P / 10
We apply separable variables:
dP / P = dt / 10
We integrate both sides:
Ln (P) = t / 10 + C
We rewrite the equation:
Exp (Ln (P)) = Exp (t / 10 + C)
P = Exp (C) * Exp (t / 10)
P = C * Exp (t / 10)
We look for the constant using:
P (0) = 300
300 = C * Exp (0/10)
300 = C * 1
C = 300
We rewrite the equation:
P = 300 * Exp (t / 10)
After 40 hours we have:
P = 300 * Exp (40/10)
P = 16379.44501
Answer:
the population size after 40 hours is:
P = 16379