2'3 Variations

1
2.3 Variations

• Variations are classes of functions in which
  there is a dependence of one physical quantity
  based on another
• These types of functions often occur when
  modeling real-world phenomenon
• Here are several types of variations where k
  represents a constant
• Direct variation
• y kx or y is directly proportional to x where
  the constant k of proporationality
• Inverse variation
• y k/x or y is inversely proportional to x (for
  k ltgt 0), we can also say that y varies inversely
  to x
• Joint variation
• z k x y, or z varies jointly as x and y (z is
  jointly proportional to x and y)

2
Direct Variation Example

• You see lightning before you hear thunder because
  light travels much faster than the speed of
  sound. The distance between you and the
  lightning varies directly as the time interval
  between the lightning and the thunder
• suppose that the thunder from a storm 5400 feet
  away takes 5 seconds to reach you determine the
  constant of proportionality and write the
  equation
• d kt or 5400 feet k 5 seconds, so k 5400
  feet / 5 seconds 1080 feet per second
• if the time interval is now 8 seconds, how far
  away is the storm?
• distance d k t 1080 feet per second 8
  seconds 8640 feet

3
Inverse Variation Example

• Boyles law states that when a sample of gas is
  compressed at a constant temperature, the
  pressure of the gas is inversely proportional to
  the volume
• a sample of air occupies 0.106 m3 at 25 degrees C
  and has a pressure of 50 kPa (a kiloPascal)
• find the constant of proportionality and write
  the equation that expresses the inverse
  proportionality
• as stated in the problem, pressure is inversely
  proportional to volume so we have P k / V
• to find k, we multiply P V 0.106 m3 50 kPa
  5.3
• if the sample expands to a volume of 0.3 m3, find
  the new pressure we use 0.3 m3 in place of 0.106
  m3
• V k / P 5.3 / .3 17.7 kPa

4
Joint Variation Example

• Newtons Law of Gravity (introduced earlier in
  the semester) determines the gravitational force
  that one object of mass m1 has on a second object
  of mass m2
• The force F is jointly proportional to their
  masses and inversely proportional to the distance
  between the objects squared, given by r
• What is the formula that relates F, m1, m2 and r
  using constant G?
• the formula F Gm1m2/r2

5
Examples Graphed
In red, we have y 3 x as x increases,
y increases proportionally by a factor of 3 In
green, we have y 3 /x as x increases, y
decreases proportionally by a factor of 3