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

2. Direct Variation Statements
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.

3. Direct Variation as nth Power
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.

4. Inverse Variation Statements
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.

5. Joint Variation Statements
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.

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

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

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