MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
A radioactive isotope decays according to dtdm=km, where m(t) is the mass (in grams) present t days after observation begins.
At t=0, the isotope has mass 120 g. After 10 days, only 82 g remain.
(a) Determine the exact value of the decay constant k.
(b) Derive an explicit formula for m(t) valid for all t≥0.
(c) Compute the half-life of the isotope, expressing your answer to the nearest hundredth of a day.
(d) Find the instantaneous rate of decay, in grams per day, when exactly 20 g of the isotope remain. Interpret the sign of your result in context.