1 precalculus i dr. claude s. moore danville community college mathematical modeling direct,...
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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°.