Algebra world problem

Question

The half-life of a radioactive substance is 31.6 years. Find the exponential decay model for this substance. How long will it take a sample of 500 grams to decay 400 grams. How much of a sample of 500 grams will remain in 20 years?

Answer 1

Hi Jaquelyn C.,

The Exponential Decay Formula is A = A_oe^-kt, where A_o is the initial amount and k is a positive constant.

If the half-life [A = (1/2)A_o] is t = 31.6 yrs then:

(1/2)A_o = A_oe^-k(31.6)

1/2 = e^-31.6k

ln(1/2) = ln(e^-31.6k)

-ln(2) = -31.6k

-ln(2)/-31.6 = k

0.022 = k

So the decay for the radioactive substance can be written as A = A₀e^-0.022t , where t is in years. You can now solve for: when A = 400g and A_o = 500g and solve for t. Also when A_o = 500g and t = 20 yr, and solve for A.

I hope this helps, Joe.