Question 1104616
not sure there were 20 incidents at first or 40.


i'll work with 40.


you started with 40 and then 3 months later there were 560.


the rate of increase for the 3 months was 560/40 = 14.


you then said, at the same rate of increase, you would multiply 40 * 14 * 14 * 14 to get 109760.


what you were doing was applying the discrete compounding formula of:


f = p * (1 + r) ^ n


f is the future value
p is the presentvalue
r is the interest rate per time period.
n is the number of time periods.


your time periods were every 3 months.


your formula would have become 560 = 40 * (1 + r) ^ 1


n was 1 because you were dealing with one 3 month period.


to find r, you would have done the following.


divide both sides of the equation by 40 to get 560/40 = (1 + r) ^ 1


since (1 + r) ^ 1 is the same as 1 + r, the equation becomes 560/40 = 1 + r.


you would then have subtracted 1 from both sides of the equation to get 560/40 - 1 = r


you would then have solved for r to get r = 13.


your formula would have become 560 = 40 * (1 + 13) ^ 1.


when n became three 3 month periods, the formula would have become f = 40 * (1 + 13) ^ 3 which would have become f = 40 * 14 ^ 3 which would have given you f = 109760.


if you used the continuous compounding formula of f = p * e^(rn) formula, then the procedure would be the same except you would have needed to do something different to get r.


f is the future value
p is the present value
r is the interest rate per time period
n is the number of time periods.


the formula for your problem would have become 560 = 40 * e^(r*1)


this is because you were dealing with one 3 month period and you were looking to find the growth rate which you could then apply to multiple three 3 month periods.


you would have solved for r as follows:


formula becomes 560 = 40 * e^(r).


divide both sides of the equaion by 40 to get 560/40 = e^r.


take the natural log of both sides of the equation to get ln(560/40) = ln(e^r).


since ln(e^r) is equal to r*ln(e) which is equal to r, you get ln(560/40) = r.


you would solve for r to get r = ln(560/40) = 2.63905733.


when n = 3,the formula would becoomes f = 40 * e^(2.63905733 * 3) which would result in f = 109760.


the continous compounding formula and the discrete compounding formula are two diferent animals and you needed to solve for the growth rate in a different way.


the formula you gave me of At=A0 * e^kt is the same as the continuous compounding formula i showed you of f = p * e^(rn).


your At was my f
your A0 was my p
your k was my r
your t was my n


different nomenclature but same formula.