Question

Element X decays radioactively with a half-life of eight minutes. If there are 800 g of element X, how long, to the nearest 10th of a minute, would it take the element to decay 22g.

Answer 1

Answer: 41.5 min

Explanation:

This problem can be solved with the Radioactive Half Life Formula:

`A=A_(o).2^{(-t)/(h)}` (1)

Where:

`A=22 g` is the final amount of the radioactive element

`A_(o)=800 g` is the initial amount of the radioactive element

`t` is the time elapsed

`h=8 min` is the half life of the radioactive element

So, we need to substitute the given values and find
`t` from (1):

`22 g=(800 g) 2^{(-t)/(8 min)}` (2)

`(22 g)/(800 g)=2^{(-t)/(8 min)}` (3)

`(11)/(400)=2^{(-t)/(8 min)}` (4)

Applying natural logarithm in both sides:

`ln((11)/(400))=ln(2^{(-t)/(8 min)})` (5)

`-3.593=-(t)/(8 min)ln(2)` (6)

Clearing
`t`:

`t=41.46 min \approx 41.5 min` This is the time elapsed