Rate-Of-Change Models In Microbiology?

Question

Here is the logistic model equation for the growth in mass of a quantity of bioremediation bacteria:
*b and o are bases of the letter m.
mb(t) = L / (1+((L-mo) / mo)e^(-Lkt)) where mb(t) is the mass of bacteria at time t, L is bounded/maximum mass, k is the growth constant, and mo is the initial mass. The model can be constructed by substituting values of mo, L and a known ordered pair for (t, mb) into the equation and solving for k.
The engineer conducting the study found that starting from an initial mass of 0.2 kg, the bacteria grow to a maximum mass of 2.6 kg following a logistic growth pattern. The mass after five days for this experiment was 1.5 kg. The engineer has modelled the mass of contaminant remaining in kilograms as
*c is the base of m and 3 is the base of log.
mc(t)= -log3((squrt of t)+1) +2.5 where mc(t) is the mass of contaminant remaining (kilograms) in t days.
1. What is the logistic growth function model for the bacterial mass?

Answer 1

I don't see where the second equation fits in but...

mb(t) = L/(1 + ((L - mo)/mo)e^(-Lkt))

You're given L (2.6), mo (0.2), and a sample point (t = 5, mb(5) = 1.5).

Plug and chug.

1.5 = 2.6/(1 + ((2.6 - 0.2)/0.2)e^(-2.6k(1.5)))

Divide both sides by 2.6 and flip both sides:

1.7333 = 1 + (2.4/0.2)e^(-3.9k)
0.7333 = 1.2e^(-3.9k)
0.6111 = e^(-3.9k)
ln 0.6111 = -0.4925 = -3.9k
k = 3.4075

So the logistic growth function is:

mb(t) = 2.6/(1 + 1.2e^(-2.6(3.4075)t))
mb(t) = 2.6/(1 + 1.2e^(-8.860t))