Answer:
319 years
Step-by-step explanation:
For a radioactive decay we have
N = N₀e^-kt , where:
N= Particles Remainng after a time t
N₀ = Particles Initially present
k = ln 2/ t₁/₂ , t₁/₂ is the half life of the radioactive element (Am-241)
We are given that the half-life is 432 yrs, from this inhformation we can calculate the value of k which then will be used to calculate the time Am-241 will take to decay to 40% once we realize we are given the ratio N/N₀.
k = ln 2/ 432 yr = 0.693/ 432 yr = 1.6 x 10⁻³ /yr
N/N₀ = e^-kt
N/N₀ = 60 (amount remaining after 40 % has decayed)/100
N/N₀ = 0.60
0.60 = e-^-kt
taking natural log to both sides of the equation to get rid of e:
ln (0.60) = -1.6 x 10⁻³ /yr x t ∴ t = - ln (0.60) /1.6 x 10⁻³/ yr
t = 0.51 /1.6 x 10⁻³ yr = 319 yrs
To verify our answer realize that what is being asked is how many years it will take to decay 40 %, and we are told the half life , 50 % decay , is 432 years, so for 40 % we will expect it will take less than that which agrees with our resul of 319 years.