Direct Variation
y varies directly as x
Graph:
y is directly proportional to x
Formula: _______________
k = constant of variation or proportionality
Examples:
d = 70 t
C = 2r
Suppose that y varies directly as x. If y is 24 when x is 8, find the constant of
variation and the direct variation equation.
Hooke’s Law states that the distance a spring stretches is directly proportional to the
weight attached to the spring. If a 50-pound weight stretches the spring 8 inches,
find the distance that an 85-lb weight stretches the spring.
Inverse Variation
y varies inversely as x
Graph:
y is inversely proportional to x
Formula: _______________
k = constant of variation or proportionality
Examples:
t = 200/r
l = 64/w
Suppose that y varies inversely as x. If y is 24 when x is 8, find the constant of
variation and the inverse variation equation.
The speed r at which one needs to drive a constant distance is inversely proportional
to the time t. A fixed distance can by driven in 5 hours at a rate of 60 mph. Find the
rate needed to drive the same distance in 4.5 hours.
Joint Variation
y varies jointly as x and z
Formula: y = kxz
Example: The area of a triangle varies jointly as its base and
its height.
The horsepower that can be safely transmitted to a shaft varies jointly as the shaft’s
angular speed of rotation (in revolutions per minute) and the cube of its diameter. A
2-inch shaft making 120 revolutions per minute safely transmits 40 horsepower.
Find how much horsepower can be safely transmitted by a 3-inch shaft making 80
revolutions per minute.
Combined Variation
[direct, inverse, joint]
The maximum weight that a rectangular beam can support varies jointly as its width
and the square of its height and inversely as its length. If a beam 4” wide, 1 foot
high, and 10 feet long can support 3 tons, find how much weight a similar beam can
support if it is 1 foot wide, 4” high, and 9 feet long.