Intermediate Algebra. 3.7 Variation. Name. Problem Set 3.7 Solutions to Every Odd-Numbered Problem. Date

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Lenard Chandler
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1 3.7 Variation 1. The variation equation is y = Kx. Substituting x = 2 and y = 10: 10 = K 2 K = 5 So y = 5x. Substituting x = 6: y = 5 6 = The variation equation is r = K s. Substituting s = 4 and r = 3:!3 = K 4 K =!12 So r =!12. Substituting s = 2: r =!12 s 2 =!6 5. The variation equation is d = Kr 2. Substituting r = 5 and d = 10: 10 = K = 25K K = 2 5 So d = 2 5 r 2. Substituting r = 10: d = 2 ( 5 10)2 = = The variation equation is y = K. Substituting x = 3 and y = 45: 2 x 45 = K = K 9 K = 405 So y = 405. Substituting x = 5: y = 405 = 405 x = The variation equation is z = Kxy 2. Substituting x = 3, y = 3, and z = 54: 54 = K 3 54 = 27K ( )( 3) 2 K = 2 So z = 2xy 2. Substituting x = 2 and y = 4: z = 2 2 ( ) 4 ( ) 2 = 64

2 11. The variation equation is I = K w 3. Substituting w = 1 2 and I = 32: 32 = K! 1$ " # 2% & 32 = K 1 / 8 K = 4 3 So I = 4 w 3. Substituting w = 1 3 : I = 4! " # 1$ 3% & 3 = 4 1 / 27 = The variation equation is z = Kyx 2. Substituting x = 3, y = 2, and z = 72: 72 = K 2 72 = 18K ( )( 3) 2 K = 4 So z = 4yx 2. Substituting x = 5 and y = 3: z = 4 3 ( ) 5 ( ) 2 = The variation equation is z = Kyx 2. Substituting x = 1, y = 5, and z = 25: 25 = K 5 25 = 5K ( )( 1) 2 K = 5 So z = 5yx 2. Substituting z = 160 and y = 8: 160 = 5 8 ( ) x = 40x 2 x 2 = 4 x = ±2

3 17. The variation equation is F = Km. Substituting F = 150, m = 240, and d = 8: 150 = K ( 240) = 240K K = 9600 K = 40 So F = 40m 40. Substituting m = 360 and d = 3: F = ( 360 ) = = The variation equation is F = Km 24 = K ( 20) = 20K 25 20K = 600 K = 30 So F = 30m. Substituting F = and m = 40: = ( ) = = 1200 = 64 d = ±8. Substituting F = 24, m = 20, and d = 5:

4 21. Let l represent the length and f represent the force. The variation equation is l = Kf. Substituting f = 5 and l = 3: 3 = K 5 K = 3 5 So l = 3 5 f. Substituting l = 10: 10 = 3 5 f 50 = 3 f f = 50 3 The force required is 50 3 pounds. 23. a. Let T represent the temperature and P represent the pressure. The variation equation is T = KP. Substituting T = 200 and P = 50: 200 = K 50 K = 4 The variation equation is T = 4P. b. Graphing the equation: c. Substituting T = 280: 280 = 4P P = 70 The pressure is 70 pounds per square inch.

5 25. Let v represent the volume and p represent the pressure. The variation equation is v = K p. Substituting p = 36 and v = 25: 25 = K 36 K = 900 The equation is v = 900. Substituting v = 75: p 75 = 900 p 75 p = 900 p = 12 The pressure is 12 pounds per square inch. 27. a. Let f represent the aperture and d represent the diameter. The variation equation is f = K d. Substituting f = 2 and d = 40: 2 = K 40 K = 80 The variation equation is f = 80 d. b. Graphing the equation: c. Substituting d = 10: f = f = 8 The f-stop is 8.

6 29. Let A represent the surface area, h represent the height, and r represent the radius. The variation equation is A = Khr. Substituting A = 94, r = 3, and h = 5: 94 = K 3 94 = 15K K = ( )( 5) The equation is A = The surface area is hr. Substituting r = 2 and h = 8: A = ( 15 8) ( 2) = square inches Let R represent the resistance, l represent the length, and d represent the diameter. The variation equation is R = Kl. Substituting R = 10, l = 100, and d = 0.01: 2 d 10 = K ( 100) 0.01 ( ) = 100K K = The equation is R = l Substituting l = 60 and d = 0.02: R = ( ) = 1.5. ( 0.02) 2 The resistance is 1.5 ohms. 33. a. Let P represent the period and L represent the length. The variation equation is P = K L. Substituting P = 2.1 and L = 100: 2.1 = K = 10K K = 0.21 The variation equation is P = 0.21 L. b. Graphing the equation:

7 c. Substituting L = 225: P = = 3.15 The period is 3.15 seconds. 35. Multiplying: 0.6( M! 70) = 0.6M! Multiplying: ( 4 x! 3) 4 x 2! 7x Simplifying: ( 4 x! 3) + 4 x 2! 7x Simplifying: 4 x 2 + 3x + 2 ( ) = 16x 3! 28x x! 12x x! 9 = 16x 3! 40x x! 9 ( ) = 4 x! x 2! 7x + 3 = 4 x 2! 3x ( ) + ( 2x 2! 5x! 6) = 4 x 2 + 3x x 2! 5x! 6 = 6x 2! 2x! Simplifying: 4 (!1) 2! 7 (!1) = 4( 1)! 7 (!1) = = 11

d = k where k is a constant of proportionality equal to the gradient.

d = k where k is a constant of proportionality equal to the gradient. VARIATION In Physics and Chemistry there are many laws where one quantity varies in some way with another quantity. We will be studying three types of variation direct, inverse and joint.. DIRECT VARIATION