A colony of bacteria is growing in a petri dish which has a maximum capacity of 140mg. The mass of bacteria is increasing at a rate given by the logistic equation. Initially there is 2mg of bacteria and the rate of increase is 1mg per day.

a. Write down the logistic equation satisfied by the mass, m. m' =
b. When will the mass of bacteria be 60mg? number of days =
c. What is the mass (in GRAMS) of bacteria 10 days after the mass was 2mg?

Step-by-step explanation:

THe logistic equation has the following form

where, in this case, P i s the mass of the Bacteria. The constant K is the maximum capacity (K = 140) and r is the growth rate (r=1) we have an initial condition of

a) The logistic equation is

In order for us to solve both questions, we must solve the differential equation, which turns out to be a separable equation. Hence

The integral on the left is solved by the method of partial fractions,

we have the following equation

or equivalently

Note that this equation most hold for all values of P. If we choose P =0, we have that A=1. If P=140, then B=1/140.

Then,

Using the substitution u = 1-P/140, we get -140du = dP, this leads to the solution

Then, the solution is given by

, where k is a constant. Note that when t=0, P=2, then

b) For solving part b, we will use the derived equation, letting P=60 and solving for t, then

. Then

. Then after 4 days the bacteria will be really close to 60 mg.

c) from the equation

with some algebra we can solve for P. This gives us

We want to know first, the amount in mg for t=10. Then

Then, P = 0.13956 Grams after 10 days.

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step-by-step explanation:

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