Differential Calculus Word Problem?
A colony of bacteria doubles in population every 20 minutes starting from an initial population size of y0. Let y(t) denote the population at time t.
1. Express y as an exponential function with base 2
2. Express y as an exponential function with base e
3. What differential equation does y satisfy?
4. At what time has the initial population grown by a factor of 3?
A colony of bacteria doubles in population every 20 minutes starting from an initial population size of y0. Let y(t) denote the population at time t.
1. Express y as an exponential function with base 2
2. Express y as an exponential function with base e
3. What differential equation does y satisfy?
4. At what time has the initial population grown by a factor of 3?
2 Answers
Explanation:
The time law describing the growth of such a population has the form
with
In term of exponential function
If we derive
Finally the population's triplication time is given by the equation
# y = y_0 2^(0.05t) # # y = y_0 e^(0.05ln2t) # # dy/dt = (0.05ln2)y # # t ~~ 32 # mins
Explanation:
1. Express y as an exponential function with base 2
2. Express y as an exponential function with base e
Take natural logarithms
3. What differential equation does y satisfy?
4. At what time has the initial population grown by a factor of 3?
We want