# Creating an Exponential decay equation for given parameters

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1. Oct 23, 2014

### shakystew

So, I am wanting to vary a parameter in an equation with respect to time.
• Vary mass flow [ m(t) ] for an exponential decay to half its original value in around 60 seconds.
I know the regular decay equation where:
m(t)=m0*exp(-At)
but I want the value to approach a steady state at 60 seconds (i.e. I am decreasing my pump/mass-flow by one half over 60 seconds).

I need an equation which will allow this to occur.

2. Oct 23, 2014

### Staff: Mentor

Is this homework?

3. Oct 23, 2014

### shakystew

It is not. It is for my current project for my research. I figured it out :)

4. Oct 24, 2014

### JJacquelin

Why not m(t) = m0 +B (exp(-A t)-1) with for example B=m0/2 or other adjusted values of A and B for better fit to the given conditions.

5. Oct 24, 2014

### HallsofIvy

Staff Emeritus
So you want m(60)= m0 exp(-60A)= (1/2)m0. Then you want exp(-60A)= 1/2 so -60A= ln(1/2), A= -ln(1/2)/60.
(Since 1/2< 1, ln(1/2)< 0 so A will be positive).
You have m(t)= m0e(-tln(1/2)/60))= m0etln((1/2)^(t/60))= m0(1/2)(t/60).

In fact, we could have argued from the first that it must be of that form- since it decreases by 1/2 every 60 seconds (exponential decay always decreases by the same fraction over the same time interval) the original amount must be multiplied by 1/2 every 60 seconds. In time t seconds, there are t/60 "60 second time intervals" so the original amount is multiplied by 1/2 t/60 times: (1/2)t/60.