Under ideal conditions, a population of rabbits has an exponential growth rate of 11.8% per day. Consider an initial population of 300 rabbits.
Find the exponential growth function.
dr/dt = .118 r
dr/r = .118 dt
ln r = .118 t + c
r = e^(.118 t +c) = e^c * e^.118t
when t = 0, e^.118t = 1
r = e^c = 300
so
r = 300 e^.118 t
To find the exponential growth function, you can use the formula:
P(t) = P0 * e^(rt)
Where:
P(t) is the population at time t
P0 is the initial population
e is the base of the natural logarithm (approximately 2.71828)
r is the growth rate (as a decimal)
t is the time (in this case, in days)
In this case, the growth rate is given as 11.8% per day, which can be converted to a decimal by dividing it by 100:
r = 11.8% / 100 = 0.118
The initial population is given as P0 = 300.
Now, you can substitute these values into the formula to find the exponential growth function:
P(t) = 300 * e^(0.118t)
So, the exponential growth function for this population of rabbits under ideal conditions is P(t) = 300 * e^(0.118t), where t is the time in days.