Question

Knowing that an unknown metal seems to decay by half every 8.75 years, what is its approximate annual decay rate?

Answer 1

Your answer is 0.571428…

Answer 2

We know exponential decay is
`A_0 = A (1-r)^t`

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'

`A_0 = A (1-r)^t`

`0.5 = 1 (1-r)^(8.75)`

`0.5 = (1-r)^(8.75)`

Divide the exponent by 8.75 on both sides

`0.5^{(1)/(8.75)} = (1-r)^{ (8.75)/(8.75)}`

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%