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DIRECT & INVERSE VARIATION

DIRECT & INVERSE VARIATION. Can you distinguish between the two?. Description. Two related quantities can vary in one of two ways: directly or inversely In direct variation, as one quantity increases, so does the other.

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DIRECT & INVERSE VARIATION

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  1. DIRECT & INVERSEVARIATION Can you distinguish between the two?

  2. Description • Two related quantities can vary in one of two ways: directly or inversely • In direct variation, as one quantity increases, so does the other. • In inverse variation, as one quantity increases, the other one decreases. • In both types, there is a constant factor k which describes the variation.

  3. Direct Variation Examples • ΔABC’s side lengths are 3 – 4 – 5. • Δ XYZ’s side lengths are 12 – 16 – 20. • The scale factor k is 4. (12 ÷ 3 = 4) • This is direct variation. • A 5” × 7” photo is enlarged so that the short side is 8”. How long is the long side? (Solve with proportion) • 5 : 7 = 8 : 11.2 (direct variation)

  4. Inverse Variation Examples • At sea level, the air pressure is 14.7 pounds per square inch. • At 9,000 feet altitude above sea level, the air pressure is 10.5 psi. • At 25,000 ft altitude, it’s 5.46 psi. • This is an example of inverse variation. • Source: www.engineeringtoolbox.com

  5. Do YOU Know Which is Which? • For each pair of quantities, state Direct if it is an example of direct variation, or Inverse if it is an example of inverse variation. • If the two quantities are not related, print “Not Related.” • Ready?

  6. How Are They Related? 1. Depth of the water vs. Water pressure • DIRECT 2. Altitude above sea level vs. Air pressure • INVERSE 3. Speed of the car vs. Time it takes to get there • INVERSE

  7. How Related? • Price of coconuts in Tahiti vs. Number of alligators in Florida • NOT RELATED • Time you spend studying for test vs. Score you make on test. • Usually it’s DIRECT, although at some point sleep will help you more

  8. How Related? • Distance from light source vs. Brightness of the light • INVERSE. Light varies inversely with the square of the distance. • Ratio of side lengths of two similar figures and ratio of their areas. • DIRECT. The area ratio is the square of the side length ratio.

  9. How Related? • Amount of taxable income vs. Amount of income tax owed. • DIRECT • Amount of paycheck deductions vs. Amount of take-home pay • INVERSE • Restaurant bill vs. Server’s tip • DIRECT

  10. How Related? • How much a photograph is enlarged vs. Picture quality • INVERSE. Picture will get grainier. • State Population vs. # of Electoral Votes • DIRECT…of course, some states are “more equal” than others

  11. How Related? • Amount of evidence against defendant vs. Likelihood of acquittal (“not guilty”) • INVERSE • Color of a car vs. How fast it goes • NOT RELATED • How much you adore your current boyfriend/girlfriend vs. How much your parents like him / her. • Usually….INVERSE

  12. Rectangular Garden • Suppose you have 400 feet of fence with which to enclose a rectangular garden. • L is length of garden, and W is the width. • The perimeter can be expressed as this equation: 2(L + W) = 400 or L + W = 200 • The area is L × W • As L increases, what happens to W? • It decreases. This is an inverse variation.

  13. Rectangular Garden • P = 400. Suppose L = 120. W = ? • L + w = 200 200 – 120 = 80 W = 80 • Area of garden is 120 × 80 = 9600 ft² • What dimensions maximize the area of the garden? • If garden is a square: L = W = 100 ft. • Area = 100 × 100 = 10,000 ft² • What dimensions make it “golden?”

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