1

PRECALCULUS I

Dr. Claude S. MooreDanville Community

College

Mathematical Modeling•Direct, inverse, joint variations;

Least squares regression

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

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

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

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

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

Writing and Using a Model

From the graph, you can see that B = 32° when r = 3.5 feet.

B = kr

32 = k3.5

112 = k

SOLUTION

Turning Radius

Ban

kin

g an

gle

(deg

rees

)

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

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 = 1125

= 22.4

Turning Radius

Ban

kin

g an

gle

(deg

rees

)

When the turning radius is 5 feet, the banking angle is about 22°.