The half-life of a radioactive substance is 27.4 years.

Question

A. Find the exponential decay model for this substance

B. How long will it take a sample of 1000 grams to decay to 600 grams

C. How much of the sample of 1000 grams will remain after 20 years

Answer 1

Tanner,

 

The amount of a radioactive substance decays at a rate which is assumed constant. The half-life is the length of time it takes a given quantity of the substance to reduce to half the original amount. This remains the same regardless of the initial amount.

 

Since radioactive decay is an example of exponential change, you can assume the amount A(t)  left after t years obeys the rule

 

A(t) = A₀e^kt,

 

where A₀ is the initial amount and k is the decay constant (which will be a negative number).

 

Since you know the half-life is 27.4 years, after this length of time any amount A₀ will reduce to A₀/2. Plug these into the formula above:

 

½A₀ = A₀e^k(27.4).

 

Solving for k,

 

1/2 = e^27.4k

 

ln(1/2) = ln (e^27.4k) = 27.4k

 

k = ln (1/2)/27.4 = − (ln 2)/27.4 ≈ −0.02530.\

 

Plug this into the function and you have,

 

A(t) = A₀e^−0.02530t.

 

This is the answer to part A.

 

B. Let A(t) = 600 g, A(t) = 600, and solve for t.

 

600 = 1000 e^−0.02530t

 

0.6 = e^−0.02530t

 

ln 0.6 = −0.02530t

 

t = (ln 0.6)/−0.02530 ≈ 20.19 years.

 

C. Using A₀ = 1000 g, and t = 20 in the same function, find A(t).

 

A(20) = 1000e^{−0.02530(20)} ≈ 602.9 g.