Question

A lab currently has 70 mg of radioactive material that decays at 3.5% per year. Federal regulation for the substance says that the material must be safely stored until it reaches its half-life, at which time the material can be disposed.

How many years will the lab have to safely

store the material before disposal?

Answer 1

The lab will need to safely store the material for approximately 19.83 years before disposal.

Step-by-step explanation:

To determine how many years the lab will have to safely store the radioactive material before disposal, we can use the concept of half-life. The half-life is the time it takes for half of the radioactive material to decay.

In this case, the material decays at a rate of 3.5% per year, so after each year, 3.5% of the material will decay.

Since we are looking for the time it takes for the amount of material to reach half of the original amount, we can use the formula:

Time = (ln(0.5) / ln(1 - 0.035))

Using a calculator, we find that the lab will need to safely store the material for approximately 19.83 years before disposal.

Answer 2

The lab will have to wait 19.45 yers to safely store the material before disposal.

Step-by-step explanation:

Exponential equation for substance decay:

A exponential equation for the amount of a 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.

Decays at 3.5% per year.

This means that
`r = 0.035`

So

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

`A(t) = A(0)(0.965)^(t)`

How many years will the lab have to safely store the material before disposal?

It needs to reach half-life, that is, t for which A(t) = 0.5A(0). So

`A(t) = A(0)(0.965)^(t)`

`0.5A(0) = A(0)(0.965)^(t)`

`(0.965)^t = 0.5`

`\log{(0.965)^t} = \log{0.5}`

`t\log{0.965} = \log{0.5}`

`t = \frac{\log{0.5}}{\log{0.965}}`

`t = 19.45`

The lab will have to wait 19.45 yers to safely store the material before disposal.