Question

The population of a bacteria culture increases at the rate of 3 times the square root of the
present population.
A. Model the population P = P(t) of the bacteria population with a differential equation.
B. Solve the differential equation that models the population P = P(t) of the bacteria
population.

C. Suppose the population at time t = 0 hours is 1000. Derive an equation for the population
P as an explicit function of time t (in hours). Your equation should contain no undetermined
constants.
D. What’s the population of the bacteria culture at the end of 10 hours? After 100 hours

Answer 1

A. dP/dt = 3SQRT(P)
B. dP = 3P^(1/2)dt
(1/3)dP*P^(-1/2) = dt
t = (2/3)P^(1/2) + c
1.5t = P^(1/2) + c
(9/4)t^2 + c = P

If we knew the population at a given time we could find out what c is, but we don't. C is just some constant.

C. 1000 = (9/4)(0) + c
c = 1000
P = (9/4)t^2 + 1000

D. P(100) = 23500 <-- just plug in 100
P(1000) = 225100 <--- plug in 1000

You may want to check that I didn't screw anything up.

Answer 2

thanks