Exponential Decay Word Question; how would I solve this?

Question

2000 tons of a radioactive element was spilled into a nearby pond. The half-life of this element is 36 days. In order for it to be declared safe for swimming there must be less than 100 tons of the material found in the pond. How long, to the nearest day, until it is safe to swim again?

Answer 1

Hello, Jake,

My solution isn't as straightforwad as Julia's, so go with her outline. I started the decay equation:

N(t) = N₀(1/2)^(t/t1/2)

where N(t) is the amount left at time t, N₀ is the initial amount, and t_1/2 is the half life (36 days in this problem)

We want to solve for the time t for when N = (1/2)N₀, or 1000.

This leads to 1000 = (2000)^(t/36)

Then solve for t. The problem is, I wasn't certain how to solve for t with putting it into Wolfram-Alpha. It provides a graph with the intersection at 155 days. I'd like to solve it directly, but this matches Julia's answer, so I'm content.

Bob

Answer 2

First we will assume this is a first order nuclear reaction, in which case we can use two equations. The first equation will help us find our decay constant, k:

t_1/2 = 0.693/k

This helps us find the decay constant using the known half life (which you gave in the question). Plug in the half life of 36 days and we should get a decay constant k of 0.01925. Knowing this, we can use the equation for first order nuclear decay:

ln(N/N_o) = -kt

where N is the current (or in this case, desired) amount of nuclear material, N_o is the initial amount of material, and k is the decay constant calculated above. We will be solving for t, the time. Plugging in the values from the problem, the equation should look like:

ln(100/2000) = -(0.01925)t

Plugging into our calculator, we get t=155.6. Our units are in days since we plugged 36 days into our first equation to solve for k. This means it will take approximately 156 days for the lake to be safe to swim in again.

We can conceptually check this number by going through chunks of 36 days and estimating the amount of material. After 36 days, there will be 1000 tons; after 72 days, 500 tons; after 108 days, 250 tons; after 144 days, 125 tons; after 180 days, 62.5 tons. It's logical that our desired tonnage would occur between 144 and 180 days.