4. The rate of decay of a radioactive substance is proportional to the amount of the substance present. Two years ago there were 5 grams of substance. Now there are 4 grams. How many grams will there be 4 years from now?
A. 16/25 |
B. 2 |
C. 64/25 |
D. 16/5 |
E. 25/4 |
Solution: Answer is C
Let S(t) = the amount of a radioactive substance at any time t.
Since the rate of decay of a radioactive substance is proportional to the amount of the substance present, we have the following differential equation,
where k is the proportionality constant and the minus represents the rate as a decaying rate. Multiply both sides of the above equation by dt and divide both sides by S(t) to obtain,
Integrating both sides yields,
where c_{1} is the constant of integration and c_{2} = e^c_{1}. Let the initial condition be the quantity of radioactive substance two years ago, that is, let S(0) = 5. So that c_{2} = 5, by substitution in the above equation. Hence,
Since there are currently 4 grams of the substance, then S(2) = 4 and we can solve for k as follows,
So,
Finally, to solve for the amount of radioactive substance four years from now, solve for S(6) as follows,