Question

A radioactive element decays according to the function q=q0e^rt where Qo is the amount of the substance at time t=0, r is the continous compound rate of decay, t is the time in years, and Q is the amount of the substance at time t. if the continous compound rate of the element per year is r=-0.00036 how long will it take a certain amount of this element to decay to half the original amount? The half-life of the element is approximately ?? Years.

Answer 1

let's say we have an initial amount of q0 = 1000

q=q0e^rt

q=1000e^rt

500=1000e^rt ... plug in q = 500 (half of 1000)

500 = 1000e^(-0.00036t) ... plug in r = -0.00036

Now the goal is to solve for t

Answer 2

500 = 1000e^(-0.00036t)

500/1000 = e^(-0.00036t)

0.5 = e^(-0.00036t)

ln( 0.5 ) = ln( e^(-0.00036t) )

ln( 0.5 ) = -0.00036t * ln( e )

ln( 0.5 ) = -0.00036t * 1

ln( 0.5 ) = -0.00036t

-0.00036t = ln( 0.5 )

t = ln( 0.5 )/(-0.00036)

t = -0.69314718/(-0.00036)

t = 1925.4088

So the half-life is roughly 1925.4088 years

To the nearest year, the half-life is roughly 1925 years