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

Answer:

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.