Question

The rate at which an amount of a radioactive substance decays is modeled by the differential equation dA/dt = kA, where A is the mass in grams, t is the time in years, and k is a constant. Answer the following.

a) If a 100-gram sample of the radioactive substance decays to 95 grams after 1 year, find an equation that can model the mass of the of the sample after t years.
b) Find the mass of the sample after 50 years.
c) The half-life of a substance is the amount of time it takes for a sample to decay to half its original size. Find the half-life of the radioactive substance.

Answer 1

let the equation be
A = 100 e^(kt)

a) if amount = 95
95 = 100 e^(1k)
.95 = e^k
k = ln .95

so A = 100 e^(ln.95 t)
when t = 50
A = 100 e^(50ln.95) = 7.69 g are left

for half-life time, only 50 g of the original 100g would remain
50 = 100 e^(ln.95 t)
.5 = e^(ln.95 t)
ln.95t = ln.5
t = ln.5/ln.95 = appr13.5 years