Applications of Exponential Models: Population and Radioactive Decay

Why Exponential Models Matter

Many quantities change at a rate that is proportional to their current size. Populations, bacteria, money in an interest-bearing account, and radioactive substances can all be modeled this way.

The key idea is simple: the larger the quantity becomes, the faster it changes. For a growing population, a population of 1,000 might add 50 individuals per year, while a population of 10,000 growing at the same proportional rate might add 500.

This leads to the differential equation

where is the quantity being modeled, is time, and is the constant proportional growth or decay rate.

The solution is an exponential function:

Here, represents the initial amount at .

Population Growth

A population that grows at a constant percentage rate is naturally modeled exponentially. The continuous-growth model is

If , the population grows. If , the population decreases.

Percentage Rate vs. Number of Individuals

This distinction is important on the AP exam. A statement such as "the population increases by 4% per year" describes a relative rate of change:

It does not mean that exactly 4 individuals are added each year. The actual number being added depends on the current population.

Worked Example 1: Modeling a Growing Population

Using the Derivative to Interpret Population Growth

The derivative of an exponential model tells you the instantaneous rate at which the population is changing.

If

then

So the rate of population growth is proportional to the population itself. This gives an important interpretation:

Radioactive Decay

Radioactive substances provide the same mathematical model, but with a negative rate constant. As radioactive material decays, the amount remaining decreases at a rate proportional to the amount currently present.

The model is

where is negative.

Because the exponent becomes increasingly negative as time passes, the quantity decreases toward zero.

Half-Life

Radioactive decay problems frequently give a substance's half-life: the amount of time required for half of the material to remain.

If the initial amount is and the half-life is , then

More generally, if the initial amount is , the amount remaining after time can be written as

This form is often easier to use than first solving for .

Worked Example 2: Radioactive Decay

Finding a Time from a Population or Decay Model

AP Calculus problems often reverse the question. Instead of asking how much remains after a certain amount of time, they ask when a quantity reaches a specified value.

Suppose

and we want to find the time when the quantity equals . Set the model equal to the target and use logarithms:

Worked Example 3: Finding When a Population Reaches a Target

Finding the Rate Constant from Two Data Points

Sometimes the growth or decay constant is not given directly. Instead, you receive two measurements. The strategy is to substitute both measurements into the exponential model and solve for the unknown constant.

For a model of the form

if the initial value is known and a later value is given, then

Taking the natural logarithm isolates .

Worked Example 4: Building a Decay Model from Data

Population Growth vs. Radioactive Decay

The mathematics is almost identical. The main difference is the sign of the constant.

• Population growth: , so the exponential function increases.
• Radioactive decay: , so the exponential function decreases.
• In both cases, the instantaneous rate of change is proportional to the current amount.
• The derivative gives the actual rate of change at a particular time.

Common Mistakes

AP Calculus Strategy

When you see a population or radioactive-decay problem, identify the mathematical structure before doing any calculations.

1. Identify the initial amount. This is the value of the model at the chosen starting time.
2. Determine whether the quantity grows or decays. This tells you whether the exponential constant should be positive or negative.
3. Write the exponential model. Use the continuous model when the problem describes continuous proportional change.
4. Use the derivative when the question asks for a rate. The model gives the amount; its derivative gives the instantaneous rate.
5. Use logarithms when solving for time or a rate constant.
6. Check the answer against the context. A decay model should decrease over time; a growth model should increase.

Practice Problems