Question

The radium in a piece of lead decomposes at a rate which is proportional to the amount present. If 10 percent of the radium decomposes in 200 years, what percent of the original amount of radium will be present in lead after 1000 years?

The answer is 59.05 - but how do I arrive at this value?

Answer 1

Assume that the decay rate in radium is proportional to some constant times the amount of radium at time t: \[dx/dt = -kx\]

We can take the derivative of this differential equation to get an expression for the amount of radium per unit time:\[x = A^{-kt} \]

Next, we solve for the decay rate k by plugging in 0.1A for amount of radium present at 200 years.
\[A*0.10=Ae^{-k*200}\]
Note that the amount of radium, x, at t = 0 is equal to A.

Solving for the decay:
\[\ln(0.10)=\ln(e^{-k*200})\]
\[-2.3=-k(200)\]
and we get k = 0.0115

Plugging back into the original equation:
\[x = Ae^{-0.0115t}\]
\[Ax = Ae^{-0.0115*1000}\]

x = some ridiculously small numer.

What have I done wrong here?

Answer 2

Hmm i know that K must be equal to 0.022 to give me the right answer of 59.05%

Answer 3

I cant seem to get that value though.

Answer 4

I solved it!

Answer 5

Can I give myself a medal?

Answer 6

good job, what did you do wrong, I couldn't find it?

Answer 7

Very elementary problem - So there is 10% decay, which means there is 90% remaining.

Answer 8

oh yeah... so you just needed the 0.9A

Answer 9

so when solving for k it should be ln(0.90) = ln(e^blahblah)

Answer 10

yea lol after all that setting up silly mistake..

Answer 11

derp, it happens to us all clearly
well I'm glad someone got it :)

Answer 12

haha me too, I'll stay on in case I run into more probs - I'll try and help out questions I see that I think i can solve