6.4 Variation 329 45. Electricity The electric current, I, in a given circuit varies inversely as the resistance, R, in the circuit. A circuit with a resistance of 8 ohms has a current of 15 amperes. Determine the current in the circuit if the resistance is 12 ohms. a) I k R = b) 10 amperes 46. Light Intensity The intensity, I, of a light source varies inversely as the square of the distance, d, from the source. If the intensity is 20 foot-candles at 15 feet, determine the intensity at 10 feet. a) I k d2 = b) 45 foot-candles 47. Distance The distance, d, that a rock falls when dropped from a cliff varies directly as the square of time, t, the rock is falling. If a rock falls 64 feet in 2 seconds, how far will it fall in 3 seconds? a) d kt2 = b) 144 ft 48. Stopping Distance of a Car The stopping distance, d, of a car after the brakes are applied varies directly as the square of the speed, s, of the car. If a car traveling at a speed of 40 miles per hour can stop in 80 feet, what is the stopping distance of a car traveling at 55 miles per hour? a) d ks2 = b) 151.25 feet 49. Wattage Rating The wattage rating of an appliance, W, varies jointly as the square of the current, I, and the resistance, R. If the wattage is 6 watts when the current is 0.2 ampere and the resistance is 150 ohms, determine the wattage when the current is 0.3 ampere and the resistance is 100 ohms. a) W kI R2 = b) 9 watts 50. Simple Interest The simple interest, i, earned in a savings account varies jointly as the amount invested, p, and the length of time of the investment, t. If $1200 invested for 3 years earns $144 of simple interest, how much simple interest does $3000 invested for 5 years earn? a) i kpt = b) $600 51. Electrical Resistance The electrical resistance of a wire, R, varies directly as its length, L, and inversely as its cross-sectional area, A. If the resistance of a wire is 0.2 ohm when the length is 200 ft and its cross-sectional area is 0.05 in. ,2 what is the resistance of a wire whose length is 5000 ft with a cross-sectional area of 0.01 in. ? 2 a) R kL A = b) 25 ohms 52. Guitar Strings The number of vibrations per second, v, of a guitar string varies directly as the square root of the tension, t, and inversely as the length of the string, l. If the number of vibrations per second is 5 when the tension is 225 kg and the length of the string is 0.60 m, determine the number of vibrations per second when the tension is 196 kg and the length of the string is 0.70 m. a) v k t l = b) 4 vibrations per second $ 53. Phone Calls The average number of daily phone calls between two cities, N, varies directly as the populations, p1 and p2, of the two cities and inversely as the square of the distance, d, between them. South Bend and West Lafayette, Indiana are 100 miles apart and the population of the two cities is 100,000 and 45,000, respectively. The average number of daily phone calls between South Bend and West Lafayette is 6750. Eugene and Corvallis, Oregon are 50 miles apart and the population of the two cities is 175,000 and 60,000 respectively. Determine the average number of daily phone calls between Eugene and Corvallis. a) N kp p d 1 2 2 = b) 63,000 calls 54. Phone Calls Refer to Exercise 53. Tuscaloosa and Auburn, Alabama are 150 miles apart and the population of the two cities is 100,000 and 78,000, respectively. Determine the average number of daily phone calls between Tuscaloosa and Auburn. a) N kp p d 1 2 2 = b) 5200 calls Concept/Writing Exercises 55. Name two items that have not been mentioned in this section that vary directly. Answers will vary. 56. Name two items that have not been mentioned in this section that vary inversely. Answers will vary. 57. a) If y varies directly as x and the constant of proportionality is 2, does x vary directly or inversely as y? Directly b) Give the new constant of proportionality as x varies directly as y. 0.5 58. a) If y varies inversely as x and the constant of proportionality is 0.3, does x vary directly or inversely as y? Inversely b) Give the new constant of proportionality as x varies inversely as y. Stays 0.3 59. a) Assume y varies directly as x. If x is doubled, how will it affect y? y will double. b) Assume y varies inversely as x. If x is doubled, how will it affect y? y will be half as large. 60. a) Assume a varies directly as b .2 If b is doubled, how will it affect a? a will be 4 times as large. b) Assume a varies inversely as b .2 If b is doubled, how will it affect a? a will be 1 4 as large. m Eugene, Oregon Hongqi Zhang/123RF Sean Pavone/Shutterstock
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