Q.The decay rate of a certain substance is directly proportional to the amount present at that instant. Initially there are 27 grams and 3 hours later 8 grams are left. The amount left after one more hour is... (A) $5\dfrac23$ grams (B) $5\dfrac13$ grams (C) $5.1$ grams (D) $5$ grams

When the mass (or amount) of a radioactive substance decreases at a rate proportional to the amount currently present, that statement translates to dx/dt = -kx with k > 0 (the negative sign records that x is decreasing), which separates and integrates to the exponential law of natural decay x = x₀·e^(-kt), where x₀ is the initial amount at t = 0. The half-life of the substance is defined as the time required for exactly half the original amount to disintegrate, i.e. the value of t at which x = x₀/2; a given half-life (or any other single before/after data point) is enough to solve for k, after which the amount at any other time — or t …