Calc AntiDerivative Help

Question

The rate of growth dP/dt of a population of bacteria is proportional to the square root of t with a constant coefficient of 6, where P is the population size and t is the time in days (0 ≤ t ≤ 10). The initial size of the population is 600. Approximate the population after 7days. Round the answer to the nearest integer.

Answer 1

To start, set up the differential equation. It is:

d P(t) / dt = 6 t^1/2

This type of differential equation can be solved by the method of separation of variables.

d P = 6 t^1/2 dt

Integrating both sides gives P = 4 t^3/2 + C where C is a constant of integration.

Since we know P(0) = 600 we see that C = 600, so the particular solution we need is thus

P(t) = 600 + 4 t^3/2

Plugging in t = 7 gives P(7) = 674.