Half life Problem which I am right and the teacher is wrong

I am dealing with Half Lives in chemistry. In case you don't know, a half life is the time it takes for half of a sample of some element to decay into other types of elements.
I want someone to check this over, review my explanations, and tell me if you either agree or disagree with me. If you disagree with me I would appreciate it if you would tell me why.

Here is the question:

Quote:

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Geologists use the decay of potassium-40 in volcanic rocks to determine their age. Potassium-40 has a half life of 1.26E^9 years, so it can be used to date very old rocks. If a sample of rock 3.15E^8 years old contains 2.73E^-7g of potassium-40 today, how much potassium-40 was originally present in the rock?

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The choices are:
a) 2.30E^-7g b) 3.25E^-7g
c) 1.17E^-8g d) 4.37E^-6



Okay, now I said 'b' for my answer, and my teacher insists it is 'a'. I don't know exactly how she got 'a' for an answer. However I will explain how I got b for an answer and how I can prove it.

First of all you can find the number of half lives by taking the total duration that you are looking at and dividing it by the half life duration.

This means the number of half lives in this problem is:
3.15E^8 years / 1.26E^9 years = .25 half lives

In order to find the amount remaining you do:
Original amount * .5 ^#halflives

This is the same as saying take the original amount, and multiply it by one half by the number of half lives that it has undergone.

But with this problem we are doing it backwards. Instead of trying to find the amount left with the original amount given we are instead given the amount left and we have to find the original amount.
This means we have to do something like:
AmountLeft * 2 ^#halflives
or
2.73E^-7g(2) ^ .25
When I do this, I get B for an answer, which is 3.25E-7g

Now, even if you didn't follow along up until I can prove my answer. Remember Original amount * .5 ^#halflives equals the amount remaining.
When you do:
3.25E-7(.5) ^ .25 you get 2.73E-7g as the amount left

Thank you to any and all who help in any way!

Today it contains t grams.
Originally it contained o grams.

This seems reasonable:
t <= o

a) and b) are therefore impossible answers.

(edit) This post is not valid anymore, due to changed numbers.

Yes. That is actually the most important, and easiest aspect to look at. I actually didn't even think of that until I made this post, LOL. I am going to have to kick my teacher in the head!

Errr... I'm not sure if you have written down the correct numbers, and the numbers and letters you keep quoting aren't matching up, but common sense says this:

The original amount has to be larger than the present amount, else it reproduced, not decayed! Second, since it's amount drops by half in approx 1x10^9 years, yet only 3x10^8 years have passed, the original amount has to be LESS than twice the present amount. Only one of the four answers you wrote down qualifies: 3.25x10^-7 grams.

Please show me what I have done wrong. I will correct it in my post. I had to make 3 edits so far fixing numbers.

And I have taken both of your points into consieration. I guess I am so used to strange numbers I forget to use COMMON SENSE to determine if the answers work :)
Thanks for replying btw

exactly true what a half life is. exacly and precisly how much time it takes for half of the material to decompose is a half life. its actually a much faster way of deciding how long it will take for the whole thing to decompose. but then again, the that wont be 100% accurate as it might actually take a little longer and decompose slower as it goes.

No biggee, you just said that 3.25 was answer B but you have it written as C - also, after seeing that answer A was less then the original, I assumed this was a misprint as well, but now I'm beginning to think not and your teacher just didn't know what she was talking about! :D

Yes. You are right. There is a name for people like me. In america we are called airheads! All that damn coffee!!

Anyway I need to make up an excuse to save some of my pride. Let me think. Oh yes, the possible answers in the book were written the way they are now (I edited it) and that is what screwed me up!

In book:
a) 2.30E^-7g b) 3.25E^-7g
c) 1.17E^-8g d) 4.37E^-6

Normal people format:
a) 2.30E^-7g
b) 3.25E^-7g
c) 1.17E^-8g
d) 4.37E^-6

OK,

The answer is 3.24654 * 10^-7

I won't post the calculations here because of the servere limitations of this forum, but here's the forumula you should use:

y = y0 * 2^(-kt)

y is the amount after t years
y0 is the original amount
k is a constant that has to be calculated (using the half-life)
t is the time in years

Yep, same that I got. With significant digits that is 'b'. We agree! Woot!

I'd love to hear your teacher's motivation on how the amount can actually increase over the years. :D :D

i dont want to think about science and math right now.

Her point was almost valid. She said something to the affect of "the element goes through a quarter half life. that means it goes though half a half life, which is .25/2 or .125"

So instead of:
2.73E^-7g(2) ^ .25

She used:
2.73E^-7(2) ^.125

And that came close to A so I guess that is where she got her answer. I don't know. She's only human, and, to tell you the truth, a wonderful teacher. She's just that: human, she makes mistakes.

Oh well.

yeah oh well. just as long as you know your right, thats all you need.