Question

A block of certain radioactive substance having mass of 140g is observed. This radioactive substance is known to decay at a rate proportional to the amount present. After 50 hours, its mass reduces to 120g, find
1. an expression for the mass of the substance at any time.
2. The time-lapse before the block decays to one quarter of its original mass.

Answer 1

\[120\div140=\frac{6}{7}\] so you can use
\[A(t)=140\times (\frac{6}{7})^{\frac{t}{50}}\] if you like

Answer 2

you are probably supposed to use
\[A(t)=140\times e^{kt}\] and find k and all that work, but it is unnecessary

Answer 3

if you want 1/4 =.25 of the original amount set
\[.25=(\frac{6}{7})^{\frac{t}{50}}\] and solve for t via
\[\ln(.25)=\frac{t}{50}\ln(\frac{6}{7})\] and do
\[t=\frac{50\ln(.25)}{\ln(\frac{6}{7})}\]