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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,