Question

The half-life of a radioactive substance is the average amount of time it takes for half of its atoms to disentegrate. Suppose you started with 200 grams of a substance with a half-life of 3 minutes. How may minutes have passed if 25 grams now remain? Explain your reasoning.

Answer 1

To solve this, we are going to use the formula:
`A=P( (1)/(2) ) ^{ (t)/(h) }`
where:

`A` the final amount of the substance

`P` is the initial amount of the substance

`t` is the time

`h` is the half life

We know from our problem that
`A=25`,
`P=200`, and
`h=3`. Lets replace those values in our formula:

`A=P( (1)/(2) ) ^{ (t)/(h) }`

`25=200( (1)/(2))^{ (t)/(3) }`

Notice that
`t` is in the exponent, so we must use logarithms to bring it down:

`(25)/(200) =( (1)/(2) ) ^{ (t)/(3) }`

`ln( (1)/(2) )^{ (t)/(3) }=ln( (25)/(200) )`

`(t)/(3) ln( (1)/(2) )=ln( (25)/(250))`

`(t)/(3)= (ln( (25)/(250) ))/(ln( (1)/(2)) )`

`t= (3ln( (25)/(250)) )/(ln( (1)/(2)) )`

`t=9`

We can conclude that 9 minutes have passed after the substance started to decay for you to have 25 gr of the substance remaining.