1. The document discusses mathematical modeling concepts including direct, inverse, and joint variation. It provides examples of how to write equations for each type of variation given data points. 2. An example is worked out showing how to find the constant of variation k given direct variation data points and then using k to write the direct variation equation. 3. The document also shows how to apply concepts of direct and inverse variation to model the banking angle of a turning bicycle based on its turning radius. The model equation is derived and used to calculate the banking angle for a given turning radius.

1. 1
PRECALCULUS I
Dr. Claude S. Moore
Danville Community College
Mathematical Modeling
•Direct, inverse, joint variations;
Least squares regression

2. 1. y varies directly as x.
2. y is directly proportional to x.
3. y = kx for some nonzero constant m.
NOTE: k is the constant of variation or the
constant of proportionality.
Example: If y = 3 when x = 2, find k.
y = kx yields 3 = m(2) or m = 1.5.
Thus, y = 1.5x.
Direct Variation Statements

3. 1. y varies directly as the nth power of x.
2. y is directly proportional to the nth
power of x.
3. y = kxn for some nonzero constant k.
NOTE: k is the constant of variation or
constant of proportionality.
Direct Variation as nth Power

4. 1. y varies inversely as x.
2. y is inversely proportional to x.
3. y = k / x for some nonzero constant k.
NOTE: k is the constant of variation or the
constant of proportionality.
Example: If y = 3 when x = 2, find k.
y = k / x yields 3 = k / 2 or k = 6.
Thus, y = 6 / x.
Inverse Variation Statements

5. 1. z varies jointly as x and y.
2. z is jointly proportional to x and y.
3. z = kxy for some nonzero constant k.
NOTE: k is the constant of variation.
Example: If z = 15 when x = 2 and y = 3,
find k.
z = kxy yields 15 = k(2)(3) or k = 15/6 = 2.5.
Thus, z = 2.5xy.
Joint Variation Statements

6. BICYCLING A bicyclist tips
the bicycle when making turn.
The angle B of the bicycle
from the vertical direction is
called the banking angle.
USING DIRECT AND INVERSE VARIATION IN REAL LIFE
Writing and Using a Model
banking angle, B

7. Writing and Using a Model
From the graph, you can see that
B = 32° when r = 3.5 feet.
B =
k
r
32 =
k
3.5
112 = k
SOLUTION
Turning Radius
Banking
angle
(degrees)
Write direct variation model.
Substitute 32 for B and 3.5 for r.
Solve for k.
Find an inverse variation model
that relates B and r.
The model is B = , where B is in degrees and r is in feet.
112
r

8. Writing and Using a Model
SOLUTION
Use the model to find the banking
angle for a turning radius of 5 feet.
Substitute 5 for r in the model you just found.
B =
112
5
= 22.4
Turning Radius
Banking
angle
(degrees
)
When the turning radius is 5 feet, the banking angle is about 22°.