Exponential Decay

Repeated Decreases by the Same Factor

Exponential decay occurs when a positive quantity is multiplied by the same factor between $0$ and $1$ during every equal interval. A discrete decay model has the form:

The quantities in this model are:

• $C>0$, the initial value
• $0<a<1$, the decay factor per interval
• $x$, the number of intervals

The factor describes the part that remains. For example, a $15\%$ decrease leaves $85\%$, so the decay factor is $a=0.85$.

Reading an Exponential Decay Graph

As $x$ increases, $a^x$ becomes smaller. The graph falls quickly at first and then flattens because each decrease is the same proportion of a progressively smaller amount.

For $f(x)=50\left(\frac12\right)^x$, the initial value is $50$ and every interval leaves half of the previous value:

The graph approaches the horizontal asymptote $y=0$ but never reaches it for any finite real $x$.

Decay Models for Medicine Radioactivity and Bounces

A decay model multiplies by a factor between zero and one at every interval. That factor removes a fixed share of what is left, so each step keeps the same proportion of the current value and the value approaches zero without reaching it.

A Half Life Model

In this simplified model, the body initially contains $50\,\mu\text{g}$ of a medicine and eliminates half of the remaining amount every hour. If $x$ is measured in hours, then:

The first values show that the model always halves the current amount:

• $f(0)=50\,\mu\text{g}$
• $f(1)=25\,\mu\text{g}$
• $f(2)=12.5\,\mu\text{g}$
• $f(3)=6.25\,\mu\text{g}$

After $9$ hours, $f(9)=50\left(\frac12\right)^9\approx0.0977\,\mu\text{g}$, which is below $0.1\,\mu\text{g}$.

Radioactive Decay

If a fraction $r$ of the radioactive nuclei decays in each time interval, the fraction of undecayed nuclei is $1-r$. For $0<r<1$:

Here, $M_0$ is the initial mass of the radioactive isotope and $M(t)$ is the mass of that isotope that has not decayed. Decay products are excluded from $M(t)$. The variable $t$ counts equal time intervals. This model assumes the stated proportional rate remains constant over the interval being studied.

Bounce Height

Suppose a ball is dropped from height $h_0$ and each rebound reaches the fixed proportion $0<p<1$ of the previous height. The height after rebound $n$ is:

For $h_0=300\text{ cm}$ and $p=\frac35$:

• $h_0=300\text{ cm}$
• $h_1=300\left(\frac35\right)=180\text{ cm}$
• $h_2=300\left(\frac35\right)^2=108\text{ cm}$
• $h_3=300\left(\frac35\right)^3=64.8\text{ cm}$

The mathematical model never reaches zero. A practical stopping statement therefore needs a threshold, such as “below $1\text{ cm}$.”

Exercises

Each problem gives an initial amount and the fraction that remains after one interval. Compute the amount that is left at the requested interval.

1. A substance starts at $200\text{ mg}$ and half of the remaining amount is eliminated every hour. How much remains after $5$ hours?

2. A radioactive sample has mass $0.3\text{ kg}$ at the start of an observation. Each hour, $15\%$ of the isotope’s nuclei decay. What mass of the undecayed isotope remains after $8$ hours?

3. A ball is dropped from $3\text{ m}$ and each rebound reaches $\frac35$ of the previous height.

   Draw the discrete rebound-height graph. Treat the ball as stopped once its rebound is below one centimetre. Find the first rebound that meets this threshold.

Worked Solutions

1. The decay factor is $\frac12$. After five hourly intervals:

   $$f(x)=200\left(\frac12\right)^x$$

   $$\begin{aligned} f(5) &=200\left(\frac12\right)^5 \\ &=200\left(\frac1{32}\right) \\ &=6.25 \end{aligned}$$

   Therefore, $6.25\text{ mg}$ remains.

2. Losing $15\%$ leaves the factor $0.85$:

   $$M(t)=0.3(0.85)^t$$

   $$M(8)=0.3(0.85)^8\approx0.08175\text{ kg}$$

   $$0.08175\text{ kg}\approx81.75\text{ g}$$

3. With height measured in centimetres, the model is:

   $$h_n=300\left(\frac35\right)^n$$
   Rebound Height

   Discrete heights with ratio $\frac35$.

   $f(x)=300\cdot(0{.}6)^x$

   $x$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$
   $f(x)$	$300$	$180$	$108$	$64{.}8$	$\approx38{.}88$	$\approx23{.}328$	$\approx13{.}9968$	$\approx8{.}39808$	$\approx5{.}038848$	$\approx3{.}0233088$	$\approx1{.}81398528$	$\approx1{.}088391168$	$\approx0{.}6530347008$

   $f(x)=300\cdot(0{.}6)^x$

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   The values near the threshold are:

   Rebound	Height in centimetres
   $n=9$	$3.023309$
   $n=10$	$1.813985$
   $n=11$	$1.088391$
   $n=12$	$0.653035$

   Rebound $11$ is still above $1\text{ cm}$, while rebound $12$ is below it. Therefore, the first rebound meeting the stated stopping rule is $n=12$.

Model Boundaries

Exponential decay is used for phenomena such as radioactive decay, depreciation, cooling approximations, and elimination models. In every application, identify the interval, the factor that remains, and the conditions under which that factor can be treated as constant. The model works while those conditions hold. If the conditions change, the pattern may also change.