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More Math Essentials

Ratios, Proportionality, and Scaling Strictly proportional Example: y = 5x Ratio gives the proportionality constant The ratio y/x = 5

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More Math Essentials

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  1. 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? More Math Essentials

  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 it’s 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. Numbers have no meaning without units We live in a linear world? y rise rise b m = run run What are units of m and b? x What about x-intercept?

  5. Ok, so it’s 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

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