We know exponential decay is [tex] A_0 = A (1-r)^t [/tex]
Where A_0 is the final value
A is the initial value
r is the decay rate
t is years
Metal seems to decay by half every 8.75 years.
Decay by half so if initial value A is 1 then final value A_0 is 0.5
t = 8.75
Now we plug in all the values and solve for 'r'
[tex] A_0 = A (1-r)^t [/tex]
[tex] 0.5 = 1 (1-r)^{8.75} [/tex]
[tex] 0.5 = (1-r)^{8.75} [/tex]
Divide the exponent by 8.75 on both sides
[tex] 0.5^{\frac{1}{8.75}} = (1-r)^{ \frac{8.75}{8.75}} [/tex]
0.923839595 = 1 - r
Subtract 1 on both sides
-0.076160404 = -r
So r= 0.076160404
Now multiply by 100 to get percentage
r = 0.076160404 * 100 = 7.61%
annual decay rate = 7.61%
