# How do you Find exponential decay rate?

##### 1 Answer

See below.

#### Explanation:

Exponential decays typically start with a differential equation of the form:

That is, the rate at which a population of something decays is directly proportional to the negative of the current population at time

We will now solve the equation to find a function of

This is the general form of the exponential decay formula and will typically have graphs that look like this:

graph{e^-x [-1.465, 3.9, -0.902, 1.782]}

**Perhaps an example might help?**

Consider a lump of plutonium 239 which initially has

We are told the lump has

Now at 1 million years:

Rearrange to get:

So

For the next part:

Rearrange to get

Now for the last part, the decay rate is already defined a way back at the very start, simply evaluate it at the given time:

The idea is to start with differential equation above, which gives the decay rate, and solve it to get the population at any given time.