A rock has 12.5% of its original amount of potassium-40 remaining in it; potassium-40 has a half-life of 1.25 billion years. How long ago was the rock formed?

Question

aaronq · Answer

do you know how to do these?

anonymous · Answer

I am horrible at these problems. I would love some help!

aaronq · Answer

i'll show you the proper way, which you can solve any odd amount of time or amount of substance. 

These are first order reactions so you can use this to find the decay constant, k:

$$t _{1/2}=\frac{ \ln2 }{ k }$$

t1/2= half life

then plug it into the exponential decay/growth formula:

$$A=A _{0}e ^{-kt}$$

Ao= initial amount
A= amount after time elapsed
t = time
k = decay constant

anonymous · Answer

What is the decay constant here?

aaronq · Answer

you find it through the first equation

anonymous · Answer

-.94

aaronq · Answer

how are you doing that?

anonymous · Answer

On my Ti-83, that's what I got for k.

aaronq · Answer

no i mean, what did you do?

anonymous · Answer

I divided 1.25 by 2, and then multipled it by LN, and multiplied that by 2.

aaronq · Answer

thats not how you solve it, 

1.25 = ln2/k

right?

isolate k,  k=ln2/1.25

aaronq · Answer

ln2 is a number, don't take it separately, it's approx 0.693

anonymous · Answer

0.47?

anonymous · Answer

Wow, I was really off. Thank you for setting me straight.

aaronq · Answer

it's 0.5545

anonymous · Answer

I didn't take it separately..

anonymous · Answer

So, how do I find ao?

aaronq · Answer

since they're giving you percentages you can take A to be 12.5
and Ao to be 100

anonymous · Answer

Okay, I will try that.

anonymous · Answer

And what is t?

aaronq · Answer

time elapsed

aaronq · Answer

what you're solving for in the problem

anonymous · Answer

Right, I am blushing ;)

aaronq · Answer

hahah oh you

anonymous · Answer

I am still unable to get a value that seems appropriate as a possible answer to this

aaronq · Answer

can you post the steps?

anonymous · Answer

12.5 = 100 e^0.5545t
Have I set it up wrong?

aaronq · Answer

12.5 = 100 e^(-0.5545t)

aaronq · Answer

you were only missing the negative sign

anonymous · Answer

Okay...if I solve the left side, I get 57.435.

aaronq · Answer

um post each step, because thats not right :S

anonymous · Answer

Perhaps it's the "e" that is causing me problems...

aaronq · Answer

do you know how to solve logarithms?

anonymous · Answer

Natural base e.

aaronq · Answer

okay, so you have:

12.5 = 100 e^(-0.5545t)

what's next?

anonymous · Answer

I calculated the right side, and i keep getting 57.43

aaronq · Answer

show me each step

anonymous · Answer

I type this: 100e^(-0.5545), which gives 57.43. ?

aaronq · Answer

you can't solve that though because there is a "t" in the exponent

anonymous · Answer

So, i have to isolate t.

aaronq · Answer

yes

anonymous · Answer

Divide 12.5 by 100e?

aaronq · Answer

just 100, e is to the exponent -0.5545t

anonymous · Answer

so 12.5/100 = e^(-0.5545)

anonymous · Answer

t

aaronq · Answer

okay, whats next?

anonymous · Answer

multiply e by -0.5545?

aaronq · Answer

you can't do that because you still have the t

anonymous · Answer

I have no idea what to do now.

aaronq · Answer

i thought you said you knew how to solve logarithms :P

aaronq · Answer

so you have this:

12.5/100 = e^(-0.5545t)

"take" ln of both sides, 

ln(12.5/100) = lne^(-0.5545t)

( you know (or you should know lol) that, lne=1 )

using logarithmic rules take the exponent to the front

ln(12.5/100) = lne^(-0.5545t)  ->  ln(12.5/100) = (-0.5545t)lne

since lne =1, so

ln(12.5/100) = (-0.5545t)(1)

then divide by -0.5545

ln(12.5/100)(-1/0.5545) = t

aaronq · Answer

do you get it?

anonymous · Answer

Yes, thank you so much. I appreciate your time and help and patience so much.

aaronq · Answer

no problem ! i hope you can do them by yourself now (not trying to be rude)