More Math Essentials

1
More Math Essentials

• Ratios, Proportionality, and Scaling
• Strictly proportional Example y 5x
• Ratio gives the proportionality constant
• The ratio y/x 5
• Scaling tells how a dependent variable changes as
  the independent variable changes. Typically, the
  independent variable is LENGTH, but in all three
  directions HEIGHT, WIDTH, and DEPTH at the same
  time.
• Example How does the volume of a bucket change
  if it expands a factor of two in all directions?

2
More Scaling

• Geometric scaling -- how do Area and Volume
  change as Length is scaled? Ratios like A/V can
  also be scaled.
• Physical scaling -- How do physical properties
  like weight, strength, heat loss, basal metabolic
  rate, etc. scale with Length (i.e., size)? First
  find which geometric quantity (L, A, V, or some
  ratio) the physical property is proportional to.
  The physical property scales the same as the
  geometric quantity.
• Examples weight strength of a rope BMR

3
Scaling is practical

• Suppose fire code sets maximum time to empty a
  stadium. The old stadium just meets this code.
  You build a new stadium, scaled up from the old
  one by a factor of 2. Will it meet the code?
• Sometimes scaling is hard, but its almost always
  important. If your customer base increases by a
  factor of 3, do you need 3 times as many
  employees? Is the magic answer to every question
  3 or 9 or 27 or something else entirely?

4
We live in a linear world?
Numbers have no meaning without units
y
rise
rise
b
m
run
run
What are units of m and b?
x
What about x-intercept?
5
Ok, so its a nonlinear world
6
More about units

• English or metric or both?
• Metric is accepted worldwide, and is easier to
  use in calculations
• Base units, like meter (m) or second (s)
• Power of 10 multipliers denoted by prefixes
• Learn these common ones.
• Doing math with powers of 10
• Multiply, you add the powers
• Divide, you subtract the powers
• Raise power to a power, multiply the powers
• Flip, you change the sign of the power
• Roots are fractional powers

7
Unit conversion

• Conversion factors are physical forms of 1
• Use conversion factor right-side-up or
  upside-down to cancel the old unit
• Multiply by conversion factor to whatever power
  is necessary to remove the old unit
• Remember to raise EVERYTHING in the conversion
  factor to the required power
• Use lots of conversion factors in a row to get
  from old unit to new unit