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 c1 is the constant of integration and c2 = e^c1. Let the initial condition be the quantity of radioactive substance two years ago, that is, let S(0) = 5. So that c2 = 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,