A colony of bacteria is growing in a petri dish which has a maximum capacity of 100mg. 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.
To understand the behavior of the bacterial growth, we can use the logistic equation, which models population growth as it approaches a maximum capacity. The logistic equation is given by:
dP/dt = r * P * (1 - P/K)
Where:
- dP/dt is the rate of change of the population (bacteria) over time.
- P is the current population size.
- r is the growth rate of the population.
- K is the carrying capacity, i.e., the maximum population size the environment can support.
In this case, we are given:
- The initial population size P0 = 2mg
- The rate of increase r = 1mg per day
- The maximum capacity K = 100mg
To find the population size at any given time, we need to solve the differential equation dP/dt = r * P * (1 - P/K) with the initial condition P(0) = P0.
To solve this equation analytically, we can separate the variables by dividing both sides of the equation by P * (1 - P/K):
(1/P) * dP/dt = r * (1 - P/K)
Next, we can integrate both sides of the equation with respect to t:
∫ (1/P) dP = ∫ r * (1 - P/K) dt
The left side can be integrated as:
ln|P| = rt + C1
Where C1 is the constant of integration.
The right side can be integrated as:
∫ (1 - P/K) dt = t - (P/K)t + C2
Where C2 is the constant of integration.
Combining the two integration results, we have:
ln|P| = rt + C1 = t - (P/K)t + C2
Since these equations hold for all t, the constants C1 and C2 must be related. We can combine them into one constant C = C1 - C2. Then, we can simplify further:
ln|P| = t - (P/K)t + C
Now, we need to solve for P. Exponentiating both sides of the equation, we get:
e^(ln|P|) = e^(t - (P/K)t + C)
|P| = e^C * e^t * e^(-(P/K)t)
Next, we can consider the absolute value of P. If P > 0, the equation becomes:
P = e^C * e^t * e^(-(P/K)t)
However, if P < 0, the equation becomes:
-P = e^C * e^t * e^(-(P/K)t)
Both cases can be combined using the absolute value function:
P = ± e^C * e^t * e^(-(P/K)t)
Finally, to find the specific solution for P, we can use the initial condition P(0) = P0:
P(0) = ± e^C * e^0 * e^(-(P0/K)*0)
2 = ± e^C * 1 * 1
2 = ± e^C
Finding the positive value of e^C, we get:
2 = e^C
Taking the natural logarithm of both sides, we find:
ln(2) = C
So the value of C is ln(2).
Now we can substitute this value back into the general solution equation:
P = e^(ln(2)) * e^t * e^(-(P/K)t)
P = 2 * e^t * e^(-(P/K)t)
This is the solution to the logistic equation describing the bacterial growth in the petri dish. By plugging in different values of t, we can find the population size at any given time.