Question

3) A sample of 42 grams of an unknown substance has a half-life of 1,300 years.

(a) Write an equation to determine the amount of substance, S, left after t years.

(b) Approximately how long will it take for 0.2 grams of substance to remain (to the nearest year)?

Answer 1

a)
`S(t) = 42(0.9995)^t`

b) It will take 10,692 years.

Explanation:

Exponential function to determine the amount of substance:

An exponential function to determine the amout of substance after t years is given by:

`A(t) = A(0)(1-r)^(t)`

In which A(0) is the initial amount and r is the decay rate, as a decimal.

(a) Write an equation to determine the amount of substance, S, left after t years.

Half-life of 1300 years means that
`A(1300) = 0.5A(0)`.

We use this to find r. So

`A(t) = A(0)(1-r)^(t)`

`0.5A(0) = A(0)(1-r)^(1300)`

`(1-r)^(1300) = 0.5`

`\sqrt[1300]{(1-r)^(1300)} = \sqrt[1300]{0.5}`

`1 - r = 0.9995`

`r = 0.0005`

Sample of 42 grams means that
`A(0) = 42`. So

`A(t) = A(0)(1-r)^(t)`

Replacing A by S, just for notation purposes

`S(t) = 42(0.9995)^t`

(b) Approximately how long will it take for 0.2 grams of substance to remain (to the nearest year)?

This is t when
`S(t) = 0.2`. So

`S(t) = 42(0.9995)^t`

`0.2 = 42(0.9995)^t`

`(0.9995)^t = (0.2)/(42)`

`\log{(0.9995)^t} = \log{(0.2)/(42)}`

`t\log{0.9995} = \log{(0.2)/(42)}`

`t = \frac{\log{(0.2)/(42)}}{\log{0.9995}}`

`t = 10692`

It will take 10,692 years.