Question 1167473
the continuous growth / decay formula is f = p * e ^ (r * t)
f is the future value
p is the present value
e is the scientific constant of 2.718281828 rounded to 9 decimal places.
r is the growth / decay rate per time period.
t is the number of time periods.


in your problem, the time period is expressed in years.


to find the rate for the half line in 5 years, do the following:


1 = 2 * e ^ (r * 5)
divide both sides of this equation by 2 to get:
.5 = e ^ (r * 5)
take the natural log of both sides of this equation to get:
ln(.5) = ln(e ^ (r * 5))
since ln(e ^ (r * 5)) = r * 5 * ln(e), and since ln(e) = 1, the formula becomes:
ln(.5) = r * 5
divide both sides of this formula by 5 to get:
ln(.5) / 5 = r
solve for r to get:
r = -.1386294361.


confirm this rate is good by replacing r in the original equation with it and solving for f.


you get:


f = e ^ (-.1386294361 * 5)
solve for f to get:
f = .5


this confirms the value of r is good.


your solution is that the continuous decay rate is equal to -.1386294361 per year.