Exponential question

Question

The half-life of a radioactive material is 12 days.  

a) Determine how much of a 5-kg sample of this material would remain after

i) 4 days ii) 3 weeks

b) When would there be 500 g of radioactive material left?

Answer 1

The "half-life" equation can be written as:

A(t) = A₀(1/2)^t/k where A(t) is the amount of material left after a period of time, A0 is the initial amount of the material, "k" is the "half life" in whatever units of time that are appropriate and "t" is the time gone by since the initial amount was measured and "t" is in the same units as "k". So, in this case:

A(t) = A₀(1/2)^t/12 where t is measured in days.

a) For A₀ = 5 kg, A(t) = 5(1/2)^t/12

i) For 4 days: A(t) = 5(1/2)^4/12 = 3.969 kg

II) For 3 weeks = 21 days, A(t) = 5(1/2)^21/12 = 1.487 kg

b) What "t" would be required for A(t) = 500 g = 0.5kg:

0.5 = 5(1/2)^t/12

0.5/5 = 0.5^t/12

0.1 = 0.5^t/12

log(0.1) = log(0.5^t/12)

log(0.1) = (t/12)•log(0.5)

log(0.1)/log(0.5) = (t/12)

t = 12log(0.1)/log(0.5) = 39.86 days