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Math Refresh • Evaluating Equations – We have 4 basic relatiojnships • • • • Y= k X Y=k/X Y=k Xn Y=k/ Xn Direct Proportion Inverse proportion Direct Exponential Inverse Exponential Direct Proportion • Y=kX • This equation of a straight line with slope k • As X increases, Y increases by same proportion – If you increase X by 10%, Y will increase 10% – If you double X, you will double Y 8 6 4 – Very straight forward 2 0 -10 -5 -2 -4 -6 -8 0 5 10 Inverse proportion • Y= k / X – If you increase X by 10%, Y will decrease by 10% – If you double X, Y will decline by factor of 2 – Y and X move in opposite direction, one increases, the other decreases 1.20 1.00 0.80 0.60 0.40 0.20 0.00 0 5 10 15 Direct Exponential • Y=kXn The Exponent n is very powerful If n=2 and you double X, then Y increases by factor of 4 If n= 3 and you double X, Y increases by factor of 9 120 100 80 60 Series1 40 20 0 0 5 10 15 Inverse Exponential • Inverse Exponential – Y=k/ Xn – Again, n is very powerful, but now it increases the effect of X in the denominator 1.20 1.00 0.80 Y 0.60 Series1 0.40 0.20 0.00 0 5 10 X 15 Math Refresher • Inverse Relationship • Y= 5/X Y declines in proportion to X increasing • Y =5/X2 Y declines much faster with X increasing ( Inverse Squared relation) • For example when x=1, then Y=5 • But when x=2, then Y = 1.25 or one fourth what is was when x =1 • What is Y when x=3? Powers of 10 • There are 31,557,600 seconds per year • Light travels at 300,000,000 meters per second • Distance traveled in one year 9.47 x 1015 meters 3.15576x 107 x 3.00 x 108 = 9.47 x 1015 meters Powers of 10 • Scale of Universe 1.5 x 1026 meters Scale of Hydrogen Nucleus 1x 10-15 meters The range of Powers of 10 from atomic scale to Scale of Universe Is captured by 41 powers of 10 Powers of 10 Rules • Multiplication – ADD the exponents Division Move the Exponent terms from denominator to numerator and change the sign of exponent Then proceed as with Multiplication, ie, add the exponents Example • E= MC2 • Solve for Mass of an electron if its mass energy is 8.19 x 10-14 joules and C = 3.0 x 108 m/s • M=E/C2 = 8.19 x 10-14 / (3.0 x 108)2 = 8.19 x 10-14 / (9.0 X 1016 ) = .91 x 10-14 x 10-16 = .91 x 10-30 = 9.1 x 10-31 kg Math Refresher • Evaluate in Powers of 10 notation • (30x 80,000,000,000)/.0004 = 3.0 x 101 x 8 x 1010 / (4 x 10-4 ) = 6 x 101 x 1010 x 104 = 6 x 1015 • 0.0001/(60,000 x 200) = 1 x 10-4 / (6 x 104 x 2 x 102 ) = .08 x 10-4 x 10-4 x 10-2 = .08 x 10-10 = 8 x 1012 Algebra • a/b = c/d Must be able to solve for any one of the four if you know the other three So: a = bc/d b = ad/c c = ad/b Math Refresher • For further review, in appendix A of Text