Direct and Inverse Variations

1
Direct and Inverse Variations

• How variables depend on each other

2
A Few Examples of Variations

• How does wind affect the velocity of a golf ball?
• How does the force with which a golf ball is hit
  affect how far it travels?
• These examples should remind you of what we
  learned about variables and experimental design!

3
Designing an Experiment

• Which golf ball travels further?
• 20 yards further on average could mean the
  difference between winning and losing a major
  LPGA tournament!
• How would you design an experiment to help
  Annika decide which golf ball to use?

Official Career Earnings 20,102,172.00
4
A Quick Review of Variables

• Independent
• Dependent
• Controlled
• Remember! The dependent variable can depend on
  many things, but in a good experiment we change
  only one (the independent variable) at a time!

5
Classifying Variables

• Qualitative variables use words to describe the
  variable
• Quantitative variables use numbers to describe
  the variable
• Remember the N!

6
Qualitative or Quantitative?

• How old are you?
• What is the speed of the car?
• What is your grade level?
• Which do you prefer, Brand 1 or Brand 2?
• How would you rate the roller coaster?
• How much time did it take?
• What size Coke do you want?
• What grade did you get on the test?

7
The Bottom Line

• A dependent variable can depend on many factors
• Volume depends on length, width and height of an
  object
• Many times we can summarize those dependences
  mathematically in an equation
• Volume length X width X height
• V lwh

8
Direct Variation

• The dependent variable increases when the
  independent variable increases
• Cost of a piece of ribbon C gets bigger as the
  length of the ribbon L gets bigger
• We can express the variation as
• CpL
• p is the proportionality constant

9
Another Example

• The volume of liquid in a graduated cylinder
  depends on the height of the liquid
• Vch

10
Idealized Case No measurement error
11
Direct Variation

• We can also have the dependence be on a power of
  the variables
• Area of a square depends on the square (second
  power) of the length of a side
• AL2 proportionality constant1
• Volume of a cube depends on the cube (third
  power) of the length of a side
• VL3 proportionality constant1
• What would a plot of A and V look like as a
  function of L? Plot it now!

12
Inverse Variation

• Just the opposite of direct variation
• If the independent variable increases, the
  dependent variable decreases
• The time to mow a field of hay depends on the
  number of mowers the farmer uses.
• T A/n Ttime n number of mowers A
  proportionality constant 10
  mower-hoursPlot the function now!