Question

Question 4 (1 point) The half life of Cs-137 is 30.2 years. If the initial mass of the sample is 100kg, how much will remain after 151 years? Leave your answer as a decimal and do not round, Answer = kg Blank 1:​

Answer 1

3.12 kg.

Explanation:

Mass after t years:

The mass of the elements after t years is given by the following equation:

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

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

The half life of Cs-137 is 30.2 years.

This means that:

`M(30.2) = 0.5M(0)`

We use this to find r.

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

`0.5 = M(0)(1-r)^(30.2)`

`(1-r)^(30.2) = 0.5`

`\sqrt[30.2]{(1-r)^(30.2)} = \sqrt[30.2]{0.5}`

`1 - r = 0.5^{(1)/(30.2)}`

`1 - r = 0.9773`

So

`M(t) = M(0)(0.9773)^(t)`

If the initial mass of the sample is 100kg, how much will remain after 151 years?

This is M(151), with M(0) = 100. So

`M(t) = 100(0.9773)^(t)`

`M(151) = 100(0.9773)^(151) = 3.12`

The answer is 3.12 kg.