A radioactive substance which starts out with 500g has decay

A radioactive substance, which starts out with 500g, has decayed to 175g after 14 days. a) Find the half-life. b) Find a function m(t) that gives the amount left after t days. c) How long will it take the substance to decay to 50g?

Solution

Initial amount = 500 gram

Amount of 14 days = 175 gram

Let us say the equation that represents the decay of the substance is:

m(t) = ae^bt

at t = 0, m = 500

This gives us a = 500

m(t) = 500e^bt

at t = 14, m = 175

175 = 500e^14b

e^14b = 0.35

14b = ln(0.35) = -1.049822

b = -0.074987

Therefore, m(t) = 500e^-0.074987t

We need to find t when m = 250

250 = 500e^-0.074987t

-0.074987t = ln(0.5) = -0.693147

t = 9.24 days

Part(b):

We already saw the equation is

m(t) = 500e^-0.074987t

Part(c)

We need to find t when m=50

50 = 500e^-0.074987t

-0.074987t = ln(0.1) = -2.3025850

t = 30.71 days