Question

A radioactive substance decays so that after t years, the amount remaining, expressed as a percent of the original amount, is A(t)= 100(1. 2)^(-t).
.
a) What is the half-life for this substance? Round to 2 decimal places.
b) Determine the rate of decay after 5 years. Round to 2 decimal places.

Many thanks!!

Answer 1

a) you want to know when the amount left is 50
50 = 100(1.2)^-t
.5 = 1.2^(-t)
log both sides
log .5 = -t log 1.2
-t = log1.2/log.5 = -.263
t = appr .263 years or appr 3.2 months

b) A'(t) = -100ln1.2(1.2)^-t
replace t with 5 and evaluate

Answer 2

0.5 = 1.2^-t
log 0.5 = -t log 1.2
-0.30 = -t * 0.0792
t =3.79 years

A = 1.2^-t
dA/dt = -t *1.2^(-t-1)
at t = 5
dA/dt =-5 * 1.2^(-6) = -5 * 0.335 = -1.67