Posted by **ronnieday** on Monday, March 19, 2012 at 12:43pm.

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.

- calculus -
**Reiny**, Monday, March 19, 2012 at 1:17pm
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

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