Exponential Decay Question

Question

A tumor is injected with 0.7 grams of Iodine-125, which has a decay rate of 1.15% per day. To the nearest day, how long will it take for half of the Iodine-125 to decay?

Answer 1

The exponential function A = P · (1 - r)^t

where A is the ending amount and P is your inital amount.

We initially start out with 100% of the iodine-125, so P will be 1

We are looking to find out how much time passes before half is remaining, so A will be 50% or 0.5

So plugging this information into our exponential function, we have:

0.5 = 1 · (1 - 0.0115)^t

0.5 = 0.9885^t

We need to solve for t which is an exponent with the base as 0.9885.

The inverse of an exponential with base 0.9885 is a logarithm with base 0.9885.

So we take the log base 0.9885 of both sides

log _0.9885 0.5 = log _0.9885 0.9885^t

The log _0.9885 and the exponent base 0.9885 cancel out on the right side

log _0.9885 0.5 = log _0.9885 0.9885^t (In other words, log _0.9885 0.9885^t = t)

log _0.9885 0.5 = t

so t = log _0.9885 0.5

Most calculators can only calculate ln (base e) or log (base 10). So we use the change of base formula.

t = log 0.5 / log 0.9885

And so finally, t = 59.9 days before only half of the iodine-125 remains.