Originally Answered: Math help on word problem, please help me solve, step by step.?
half life is defined as the time it takes for 1/2 the material to decay. For material such as plutonium, the decay rate is proportional to the amount of substance , so more the material faster it decays. However the rate slows down as the material keeps decaying
so our logic is that in every 24360 years half the plutonium remaining decays away.
like, 100 grams becomes 50 grams in 24360 years, 50 becomes 25 after another 24360 years and so on.
to use algebra , let us consider 24360 years as a period. In one period (p=1)
in 2 periods (p=2) 50/2=100/4=25 and so on
So in n periods w=100/2^n
lets assume this formula for all values of n
let weight=75g in x periods
taking log on both sides
no of years = 0.415*24360=10109 years
the justification to use the equation w=100/2^n answer can really only be found using calculus
(sovling the differential equation dw/dt=-kw where k is a constant), yielding
w is the initial quantity of the substance that will decay (this quantity may be measured in grams, moles, number of atoms, etc.),
w(t) is the quantity that still remains and has not yet decayed after a time t,
t1 / 2 is the half-life of the decaying quantity,
τ is a positive number called the mean lifetime of the decaying quantity,
λ is a positive number called the decay constant of the decaying quantity.
The three parameters t1 / 2, τ, and λ are all directly related in the following way:
half life t1/2= ln(2)/λ=τ*ln(2)
here half life = 24360=ln(2)/λ
now using this equation to compare weight 75=100e^-λt
take natural log on both sides
Yields pretty much the same answer. Must be remembered that both the equations are pretty much the same.