# Decay rates in exponential decay

• Dec 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?
• Dec 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, $\displaystyle \alpha$ does not become $\displaystyle 0.01$, rather the product of $\displaystyle \alpha$ and $\displaystyle x^{1000}$ becomes $\displaystyle 0.01$

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

to solve for $\displaystyle x$ ...

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

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

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

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

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

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

$\displaystyle x = 0.996095619...$
• Dec 21st 2009, 06:20 AM
Soroban
Hello, misterprogrammer!

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, $\displaystyle x$ to repeatedly multiply my initial amount $\displaystyle A_o$
so that over 1000 steps $\displaystyle A_o$ 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: .$\displaystyle A \:=\:A_ox^t$

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

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

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