Decay rates in exponential decay

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December 21st 2009, 04:54 AM
misterprogrammer

Decay rates in exponential decay

Hi there, I'm actually trying to write a little computer program which will invlove me using a half-life type of decay, and I'm not really sure how to go about various calculations.

What I want to calculate is a particular value, x, to repeatedly multiply my decaying variable Alpha by (Alpha is given and can be between 0.01 and 1), so that over 1000 steps Alpha becomes 0.01.

Now, I did this by hand for an initial alpha of 1 and got a value 0.995405418.

I'm thinking that could possibly boil down to

0.01 = Alpha ( x ^ 1000)

But I don't have the math skills to rearrange for x, and I'm not sure that's the right answer anyway.

Any pointers?
December 21st 2009, 06:17 AM
skeeter

Quote:

Originally Posted by misterprogrammer

Hi there, I'm actually trying to write a little computer program which will invlove me using a half-life type of decay, and I'm not really sure how to go about various calculations.

What I want to calculate is a particular value, x, to repeatedly multiply my decaying variable Alpha by (Alpha is given and can be between 0.01 and 1), so that over 1000 steps Alpha becomes 0.01.

Now, I did this by hand for an initial alpha of 1 and got a value 0.995405418.

I'm thinking that could possibly boil down to

0.01 = Alpha ( x ^ 1000)

But I don't have the math skills to rearrange for x, and I'm not sure that's the right answer anyway.

Any pointers?

the way you have set up your equation, $\alpha$ does not become $0.01$ , rather the product of $\alpha$ and $x^{1000}$ becomes $0.01$

it sounds like you want $\alpha$ to be a given fixed constant $0.01 \le \alpha \le 1$

to solve for $x$ ...

$0.01 = \alpha \cdot x^{1000}$

$\frac{0.01}{\alpha} = x^{1000}$

$\left(\frac{0.01}{\alpha}\right)^{\frac{1}{1000}} = x$

so ... say the given $\alpha = 0.5$

$\left(\frac{0.01}{0.5}\right)^{\frac{1}{1000}} = x$

$(0.02)^{\frac{1}{1000}} = x$

$x = 0.996095619...$
December 21st 2009, 06:20 AM
Soroban

Hello, misterprogrammer!

Your description is off.
I hope I've interpreted your intentions correctly.

Quote:

I'm trying to write a computer program which invloves a half-life type of decay.

What I want to calculate is a particular value, $x$ to repeatedly multiply my initial amount $A_o$
so that over 1000 steps $A_o$ is reduced by a factor of 0.01

Now, I did this by hand for an initial alpha of 1 and got a value 0.995405418.
. . This is correct!

We have: . $A \:=\:A_ox^t$

When $t = 1000$ , we want $A$ to be 0.01 of $A_o$

So, we have: . $0.01A_o \:=\:A_ox^{1000} \quad\Rightarrow\quad x^{1000} \:=\:0.01$

Therefore: . $x \;=\;(0.01)^{\frac{1}{1000}} \;=\;0.995405417$