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Understanding Exponential Decay

Exponential Decay is used to determine how long an element may exist before hitting a certain threshold. Radioative material, for instance, has a "half-life" (the time before half of its radioactivity dissipates), and this is used to determine when the material is safe for handling. Another instance of exponential decay is the amount of time a drug remains in your body - for example, if you are hospitalized, doctors and nurses need to know when to give you additional medicine as to keep the medicine working in you, but not to give you an overdose!

Exponential Decay Formula
A = Aoekt

Where:
A = Present-time mass/concentration
Ao = Mass/concentration that we are looking for.
e = log function
k =
t = time span

Example
After 5 hours, a medicine has decayed to 72.4% of its original concentration. Find the half-life of the medicine.

Using our formula, we plug in the values we know:

Present concentration = A = 72.4. Convert this to numeric instead of percentage: A = .724 (This means t = 0, which is the same as t = "5 hours ago."

We are told that after 5 hours, the concentration is 72.4%. We convert this and plug the numbers into our formula:
When t = 5, A=.724 Ao. Thus, our formula to determine 'k' is:

.724Ao = Aoek(5)

Divide both sides by Ao: .724 = ek(5)

Take the natural log of both sides to remove the e:

LN (.724) = ln(ek(5))

This equates to: LN(.724) = k(5) or LN(.724) = 5k

Now we want to get 'k' by itself, so we divide again: k = ln(.724) / 5

And that makes k = -0.065.

Since we now have a solid value for 'k', we need to put it back into the original formula:

A = Aoe(-.065)t

And we find the half-life. The half-life, is like it says, half, so we multiply by 1/2:

(1/2)Ao = Aoe(-.065)t

Again, we divide both sides by Ao to eliminate it:

1/2 = e(-.065)t

Solve for t by eliminating e via the natural log function:

LN(1/2) = (-.065)t

And finally, we get 't' by itself by dividing both sides by -.065:

t = LN(1/2) / (-.065)

Using a calculator, we determine that t = 10.66.

That is, the medicine will be at 50% of its original concentration after 10.66 (nearly 11) hours inside the body!

 

As with Exponential Growth, Exponential Decay is only made difficult by the multiple steps needed to take to get the result to answer the original question - in our example, we basically had to do two problems! The math portion, however, is straightforward and simple, so now, instead of using deep knowledge of mathematics, you are using thinking and logic skills. Those logic skills allow you to determine what your answer means, and how to use it to get the answer to the question, if necessary!

 

Find another math lesson at the Math Lesson and Tutorial Center!

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