Notes 9B Linear Modeling
In Unit 9A, we represented functions with tables and graphs. We now turn our attention
to a more common and versatile way of representing functions: with equations. Although
equations are more abstract than pictures, they are easier to manipulate mathematically
and give us greater power when creating and analyzing mathematical models. We can
understand the basic principles of mathematical modeling by focusing on the
simplest models: linear models, which can be represented by linear functions, meaning
functions that have straight-line graphs.
Example 1 Drawing a Linear Model
You have a 3-mile trail, starting at an elevation of 8,000 feet. Along the way, the
trail gains elevation at a rate of 650 feet per mile. What is the domain for the elevation
function? From the given data, draw a graph of a linear function that could represent a
model of your elevation as you hike along the trail. Does this model seem realistic?
Example 2 A Price-Demand Function
A small store sells fresh pineapples. Based on data for pineapple prices between
$2 and $7, the storeowners created a model in which a linear function is used to describe
how the demand (number of pineapples sold per day) varies with the price.
1. What does the point (2,80) mean?
2. What is the rate of change for this function?
3. Discuss the validity of this model.
Example 3 Change in Demand
Using the linear demand function in previous figure, predict the change in demand for
pineapples if the price increases by $3.
Also known as y = mx + b where m is your slope and b is your y-intercept
Example 4 Alcohol Metabolism
Alcohol is metabolized by the body (through enzymes in the liver) in such a way that the
blood alcohol content decreases linearly. A study by the National Institute on Alcohol
Abuse and Alcoholism showed that, for a group of fasting males who consumed four
drinks rapidly, the blood alcohol content rose to a maximum of 0.08 g/100 mL about an
hour after the drinks were consumed. Three hours later, the blood alcohol content had
decreased to 0.04 g/100 mL. Find a linear model that describes the elimination of
alcohol after the peak blood alcohol content is reached. According to the model, what is
the blood alcohol content five hours after the peak is reached?
Example 5 Price from demand
Write an equation for the linear demand function of the pineapple problem. Then
determine the price that should result n a demand of 80 pineapples per day.
Example 6 Crude Oil Use Since 1850
Until about 1850, humans used so little crude oil that we can call the amount zero – at
least in comparison to the amount used since that time. By 1960, humans had used a total
(cumulative) of 600 billion cubic meters of oil. Create a linear model that describes
world oil use since 1850. Discuss the validity of the model.