HW 6.7 #11
Find the half-life of an element which decays by 3.418% each day. Hint: use y = ab^t.

First setup the equation for how much element is remaining after each day:

 y = a ( 100% - 3.418%)^t

where y is the amount remainig and t is the number of days

We can clean this up by subtracting the percentages and converting that to a decimal:

  y = a (0.96582)^t

Now for the half-life, we need to figure out how long until there's half of what we started with.  When does "a (0.96582)^t" equal "1/2 of a"

1/2a = a (0.96582)^t

divide by a on both sides and you get

1/2  = (0.96582)^t

take the log of both sides and you get:

log(1/2) = t log(0.96582)

divide both sides by log(0.96582)

 log(1/2) / log(0.96582) = t

Throw that into a calculator and you have

 t ≈ 19.93