Exponential Growth and Decay; Modeling Data

Exponential Growth and Decay Models

Exponential functions are used to model situations where quantities grow or decay at rates proportional to their current value. These models are fundamental in Precalculus for understanding population dynamics, radioactive decay, and other real-world phenomena.

• Exponential Growth Model: The function models the amount of a growing entity, where:

  • is the initial amount at time

  • is the amount at time

  • is the growth rate constant ()

• Exponential Decay Model: The function models the amount of a decaying entity, where:

  • is the initial amount at time

  • is the amount at time

  • is the decay rate constant ()

Example: Population Growth

Suppose the population of Africa was 807 million in 2000 and 1341 million in 2020. Using the exponential growth model, we can find the function that describes this growth:

• Let be the number of years after 2000.

• Model:

• Using the data, solve for and write the function.

• Application: By the year 2036, Africa’s population will reach 2000 million.

Example: Radioactive Decay (Strontium-90)

The half-life of strontium-90 is 28 years. The exponential decay model is:

• Half-life means

• Solve for using the half-life formula.

• Application: If there are originally 60 grams, it will take about 72 years for strontium-90 to decay to 10 grams.

Logistic Growth Model

Logistic growth models describe situations where growth is limited by resources, leading to a maximum carrying capacity. The general form is:

• where , , and are constants and is the limiting value.

Example: Learning Theory

• Model: describes the proportion of correct responses after learning trials.

• Prior to learning trials, the proportion is 0.4.

• After 10 trials, the proportion is 0.7.

• The limiting proportion as is 0.8.

Newton’s Law of Cooling

Newton’s Law of Cooling models the temperature change of an object placed in a cooler environment. The temperature at time is:

• is the constant temperature of the surroundings

• is the initial temperature of the object

• is a negative constant

Example: Cooling Object

• An object is heated and left to cool in a room.

• Model:

• After 20 minutes, substitute to find the temperature.

• To find when the object reaches a certain temperature, solve for .

Choosing a Model for Data

When modeling real-world data, it is important to choose the function type that best fits the data. Scatter plots can help visualize the relationship.

• Example: Table of city populations and average walking speeds.

• Scatter plot suggests a logarithmic function is a good choice for modeling the data.

Expressing Exponential Models in Base e

Any exponential model can be rewritten in terms of base :

• is equivalent to

• This is useful for expressing models in terms of natural logarithms.

Example: Rewriting a Model

• Rewrite in terms of base :

• Round to three decimal places if needed.