Question 1191731
one of the formula you can use is:
f = p * (1 + r) ^ n
when f = 1 and p = 2 and n = 5.27, you get:
1 = 2 * (1 + r) ^ 5.27
divide both sides of this equation by 2 to get:
.5 = (1 + r) ^ 5.27
take the 5.27th root of both sides of this equation to get:
.5 ^ (1/5.27) = (1 + r)
solve for (1 + r) to get:
(1 + r) = .8767556206.
that's your growth factor per year.


when p = 150 and f = 20, the formula becomes:
20 = 150 * .8767556206 ^ n
divide both sides of this equation by 150 to get:
20/150 = .8767556206 ^ n
take the log of both sides of this equation to get:
log(20/150) = n * log(.8767556206).
divide both sides by log(.8767556206) to get:
log(20/150) / log(.8767556206) = n
solve for n to get:
n = 15.31931344.


.8767556206 and 15.31931344 are rounded to the number of digits that the calculator can display.
i stored these number into memory, which take the answer out to more decimal digits.
this makes it more accurate.
i used the stored number to confirm the answer is correct, even though i am showing you the displayed numbers


2 * .8667556206 ^ 5.27 = 1
1 is 1/2 of 2, so that's your half life.
the growth factor per year is .8667556206.
2 will shrink to 1 in 5.27 years at a yearly growth rate of .8667556206.


150 * .8667556206 ^ 15.31931344 = 20.
150 will shrink to 20 in 15.31931344 years at a yearly growth rate of .8667556206.


note that (1 + r) = .8667556206.
solve for r to get:
r = .8667556206 - 1 = -.1232443794.
that's your annual growth rate.
the life of cobalt -60 is being reduced by approximately 12.32% each year.


the equation can be graphed.
it looks like this.


<img src = "http://theo.x10hosting.com/2022/030603.jpg" >


in the graph, you can see that the remaining life of cobalt -60 if 150 years at the beginning, then 75 after 5.27 years, then 20 after 15.3931344 years.