MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
Carbon-14 decays radioactively over time. A physicist records the following statement regarding a pure sample of Carbon-14:
“The rate at which the mass decreases at any instant is proportional to the mass remaining at that instant.”
Let M(t) represent the mass (in milligrams) of the sample t years after the beginning of the observation. The half-life of Carbon-14 is known to be approximately 5,730 years.
(1) Write a differential equation that models the decay of Carbon-14 described above.
(2) Derive an explicit formula for M(t) that satisfies the differential equation, expressing your answer in terms of an undetermined decay constant k.
(3) Using the given half-life, calculate the exact value of k (express the answer as an exact real number or a decimal rounded to five significant figures).
(4) Determine the percentage of the original mass remaining after 20,000 years. Express your answer to the nearest tenth of a percent.
(5) Assume that 3.0 mg of Carbon-14 remain in an archaeological sample today. Estimate the mass of Carbon-14 that was present in the sample 10,000 years ago, according to your model. Show all work.