Chapter Five - New AMS and AWM Fellows | LSU Math
Section 5.6 Applications of Exponential and Logarithmic Functions
Objectives
1. Solving Compound Interest Applications
2. Solving Exponential Growth and Decay Applications
We have seen that exponential functions appear in a wide variety of settings including biology, chemistry, physics and business.
Objective 1: Solving Compound Interest Applications
In Section 5.1 and Section 5.2, the formula for compound interest and continuous compound interest were defined as follows:
Compound Interest Formulas
Periodic Compound Interest Formula
[pic]
Continuous Compound Interest Formula
[pic]
[pic]Total amount after t years
[pic]Principal (original investment)
[pic]Interest rate per year
[pic]Number of times interest is compounded per year
[pic]Number of years
Objective 2: Exponential Growth and Decay Applications
The exponential growth model is used when a population grows at a rate proportional to the size of its current population. This model is often called the uninhibited growth model.
Exponential Growth
The graph of
A model that describes the exponential uninhibited growth of a [pic]
population, P, after a certain time, t, is for [pic]
[pic]
where [pic]is the initial population and [pic]is a constant
called the relative growth rate.(Note: [pic]is sometimes given as a percent.)
Exponential Decay
Some populations exhibit negative exponential growth. In other words, the population, quantity or amount decreases over time. Such models are called exponential decay models. The only difference between an exponential growth model and an exponential decay model is that the constant, k, is less than zero.
Exponential Decay
A model that describes the exponential decay of a The graph of
population, quantity or amount A, after a certain time, t, is [pic]
for [pic]
[pic]
where [pic]is the initial quantity and [pic]is a constant
called the relative decay constant. (Note: [pic]is sometimes given
as a percent.)
Half-Life
Every radioactive element has a half-life, which is the required time for a given quantity of that element to decay to half of its original mass. For example, the half-life of Cesium-137 is 30 years. Thus it takes 30 years for any amount of Cesium-137 to decay to ½ of its original mass. It takes an additional 30 years to decay to ¼ of its original mass and so on.
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