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6.8 Solving (Rearranging) Formulas & Types of Variation Rearranging formulas containing rational expressions Variation Variation Inverse Joint Combined 6.8 1 Electronics: “Solving” formulas for different variable Solve the electronic resistance formula for the variable r1 What’s the LCD? 1 1 1 R r1 r2 1 1 1 Rr1r2 R r1 r2 r1r2 Rr2 Rr1 r1r2 Rr1 Rr2 r1 r2 R Rr2 Rr2 r1 r2 R 6.8 2 Astronomy: “Solving” formulas for different variable Solve for the height variable h in the satellite escape velocity equation What’s the LCD? V2 2g R2 R h V 2 2g 2 2 R R h Rh R V 2 R h 2 gR2 V 2 R V 2 h 2 gR2 V 2 h 2 gR2 V 2 R 2 gR2 V 2 R h V2 6.8 3 Acoustics (the Doppler Effect): “Solving” formulas for different variable Solve for the speed variable s in the doppler effect equation What’s the LCD? sg f sv sg f s v sv f s v sg fs fv sg fs sg fv s( f g ) fv fv fv s f g g f 6.8 4 Direct Variation The words “y varies directly with x” or “y is directly proportional to x” mean that y = kx for some nonzero constant k The constant k is called the constant of variation or the constant of proportionality Express the verbal model in symbols: “A varies directly with the square of p”. A = kp2 Find the constant of variation, if A = 18 when p = 3 18 = k(3)2 so 18 = 9k therefore k = 2 Real: Distance of a lightning bolt varies directly with the time 6.8 between seeing the flash and hearing the thunder. m = (1/5)s 5 Inverse Variation The words “y varies inversely with x” or “y is inversely proportional to x” mean that y = k/x for some nonzero constant k The constant k is called the constant of variation Express the verbal model in symbols: “z varies inversely with the cube of t”. z = k/t3 Find the constant of variation, if t = 2 when z = 10 10 = k/23 so 10 = k/8 therefore k = 80 Real: Loudness of sound varies inversely with the square of the distance from the sound. L6.8= k/d2 6 Joint Variation The words “y varies jointly with x and z” or “y is jointly proportional to x and z” mean that y = kxz for some nonzero constant k The constant k is called the constant of variation Express the verbal model in symbols: “M varies inversely with the cube of n and jointly with x and the square of z”. M = kxz2/n3 Find the constant of variation, if M = 3 when z=10, x=2, n=1 3 = k(2)(10)2/13 so 3 = 200k therefore k = 3/200 6.8 7 Solving Variation Problems (at least two sets of values) 1. Translate the verbal model into an equation. 2. Substitute the first set of values into the equation from step 1 to determine the value of k. 3. Substitute the value of k into the equation from step 1. 4. Substitute the remaining set of values into the equation from step 3 and solve for the unknown. ELECTRONICS The power (in watts) lost in a resistor (in the form of heat) is directly proportional to the square of the current (in amperes) passing through it. The constant of proportionality is the resistance (in ohms). What power is lost in a 5-ohm 6.8 resistor carrying a 3-ampere current? w kc2 w 5(3) 2 w 5(9) w 45 45 watts of lost power 8 Heating up the Gas (mixed variation) The pressure of a certain amount of gas is directly proportional to the temperature (measured in degrees Kelvin) and inversely proportional to the volume. A sample of gas at a pressure of 1 atmosphere occupies a volume of 1 cubic meter at a temperature of 273° Kelvin. When heated, the gas expands to twice its volume, but the pressure remains constant. To what temperature is it heated? 6.8 T P k first , find k V 273 1 1 k so k 1 273 1 T 1 273 2 546 T The gas was heated to 546 K 9 What Next? Exponents and Radicals - Section 7.1 6.8 10