The formula for exponential growth is expressed as
y = yo e^kt
where
yo is the initial population
y is the final population after time, t
t is the duration
when y = 500, t = 15
500 = yo e^15k
when y = 500, t = 15
1700 = yo e^40k
Dividing equation 2 by equation 1, we have
1700/500 = e^40k/e^15k
3.4 = e^(40k - 15k) = e^25k
Taking natural log of both sides
ln 3.4 = ln e^25k
25k = ln 3.4
k = ln 3.4/25
k = 0.049
The equation would be
y = yo e^0.049t
500 = yo e^15 *0.049
500 = yo e^0.735
yo = 500/e0.735
yo = 239.75
Doubling time is when y = 2yo Thus, we have
2yo = yo e^0.049t
2 * 239.75 = 239.75e^0.049t
479.5/239.75 = e^0.049t
2 = e^0.049t
Taking natural log of both sides
ln 2 = lne^0.049t
0.049t = ln2
t = ln2/0.049
t = 14.16
Doubling time = 14.16 minutes
For 60 minutes, t = 60
y = 239.75e^0.049*60
y = 4535.07
For y = 12000, we have
12000 = 239.75e^0.049t
12000/239.75 = e^0.049t
50.05 = e^0.049t
Taking natural log of both sides
ln 50.055 = lne^0.049t
0.049t = ln50.05
t = ln50.05/0.049
t = 79.86 minutes
