Question

Element X decays radioactively with a half life of 12 minutes. If there are 200 grams of Element X, how long, to the nearest tenth of a minute, would it take the element to decay to 50 grams?

Answer 1

It would take 24 minutes for the element to decay to 50 grams

Explanation:

The equation for the amount of the element present, after t minutes, is:

`Q(t) = Q(0)e^(-rt)`

In which Q(X) decays radioactively with a half life of 12 minutes.(0) is the initial amount and r is the rate it decreases.

Half life of 12 minutes

This means that
`Q(12) = 0.5Q(0)`

So

`Q(t) = Q(0)e^(-rt)`

`0.5Q(0) = Q(0)e^(-12r)`

`e^(-12r) = 0.5`

`\ln{e^(-12r)} = ln(0.5)`

`-12r = ln(0.5)`

`12r = -ln(0.5)`

`r = -(ln(0.5))/(12)`

`r = 0.05776`

If there are 200 grams of Element X, how long, to the nearest tenth of a minute, would it take the element to decay to 50 grams?

This is t when Q(t) = 50. Q(0) = 200.

`Q(t) = Q(0)e^(-rt)`

`50 = 200e^(-0.05776t)`

`e^(-0.05776t) = 0.25`

`\ln{e^(-0.05776t)} = ln(0.25)`

`-0.05776t = ln(0.25)`

`0.05776t = -ln(0.25)`

`t = -(ln(0.25))/(0.05776)`

`t = 24`

It would take 24 minutes for the element to decay to 50 grams