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Exponential Decay

Exponential decay models apply to any situation where the decay (decrease) is proportional to the current size of the quantity of interest.  Such situations are encountered in biology, business, chemistry and the social sciences.

Exponential decay models are also used very commonly, especially for radioactive decay, drug concentration in the bloodstream, of depreciation of value.

Radioactive Decay

Radioactive decay problems are often given in terms of half-life of a radioactive element. This is modeled by the equation:

$N=N_0{2}^{-\left(t ÷ \tau \right)}$

where $N_0$ is the initial amount of the element, $N$ is the amount remaining after $t$ years, and $\tau$ is the half-life.

Here, $N_0=2000$ and $\tau =\mathrm{1,260,000,000}$ . So the model is:

$N=2000\cdot 2^{-\left(t ÷ 1260000000\right)}$

Drug Concentration

For drug concentration problems, you may be given the fraction $p$ of the original amount of the drug left in the bloodstream after a unit of time. In this case, the situation is modeled by the equation

$y=A{p}^t$ ,

where $y$ is the concentration remaining after time $t$ , and $A$ is the initial amount .

Substitute.

$y=200{\left(0.8\right)}^4$

You might want a calculator!

$\begin{array}{l}y=200\left(0.4096\right)\\ y=81.92\end{array}$

So there are about $82$ milligrams of the drug left in the bloodstream after four hours.

Depreciation

If the value of some article (for example, a car), originally $\$C$ , depreciates $x\%$ per year, then the value after $t$ years is given by the formula:

$y=C{\left(1-\frac{x}{100}\right)}^t$

Substitute.

$\begin{array}{l}y=28000{\left(1-0.15\right)}^4\\ =28000{\left(0.85\right)}^4\\ =28000\left(0.52200625\right)\\ =14616.175\end{array}$

So after four years, the car is worth about $\$\mathrm{14,616}$ .