Direct and inverse proportion

Transcription

1 CHAPTER 34 Direct and inverse proportion 34 CHAPTER Direct and inverse proportion 34.1 Direct proportion When one quantit increases in the same proportion as another quantit, the quantities are said to be directl proportional to each other (see Section 20.4). For eample, the cost of a bag of potatoes is directl proportional to the weight of the potatoes. The smbol is used to denote direct proportion. If the cost of the bag of potatoes is C pence and the weight of the potatoes is W kg then C W. If the potatoes cost k pence per kilogram then C kw. In general if is directl proportional to and k where k is a number known as the constant of proportionalit. Since k the graph of against is a straight line passing through the origin. k O The constant of proportionalit, k, is the gradient of this straight line. Eample 1 is directl proportional to. When 30, 45 Find when 40 Solution 1 so k 45 k 30 k Substitute 30 and 45 Find k This is the formula for in terms of Substitute 40 into

2 34.1 Direct proportion CHAPTER 34 Eample 2 The voltage, V volts, across an electrical circuit is directl proportional to the current, I amps, flowing through the circuit. When I 1.2, V 78 a Epress V in terms of I. b Find V when I 2 c Find I when V Solution 2 a V I so V ki 78 k k V 65I b V Substitute V 78 and I 1.2 Find k Substitute I 2 into V 65I c I I Substitute V into V 65I Eercise 34A 1 is directl proportional to. 2 is directl proportional to. a 10 when 2 Find when 3 a 8 when 2 Find when 10 b 5 when 3 Find when 4.5 b 6 when 4 Find when 7.5 c 6 when 2 Find when 3.3 c 7 when 2 Find when 3 d 3 when 8 Find when 6 d 8 when 5 Find when 13 3 The height, T mm, of a pile of paper is 4 The distance, D km, travelled b a car is directl proportional to the number of directl proportional to the amount, sheets, N, in the pile. A litres, of petrol used. When N 250, T 28 When A 5, D 270 a Find a formula for T in terms of N. a Epress D in terms of A. b Find the value of T when N 300 b Find the value of D when A 4.5 c Find the value of N when T 98 c How man litres of petrol are needed for a journe of 324 km? 5 The time, T seconds, taken for a pan of water to boil is directl proportional to the amount, A litres, of water in the pan. When A 2.4, T 150 a Find a formula for T in terms of A. b Find the value of T when A 1.8 c If the pan takes 3 minutes to boil, how much water is in it? 6 The perimeter, P cm, of a regular heagon is proportional to the length, l cm, of its longest diagonal. When l 3.6, P 10.8 a Find a formula for P in terms of l b The value of l increases from 4.8 to 5.4 Find the increase in the value of P. c The value of l increases b 20%. Find the percentage increase in the value of P. 551

3 CHAPTER 34 Direct and inverse proportion 34.2 Further direct proportion Sometimes one quantit is directl proportional to the square or the cube of another quantit. For eample, the area, A cm 2, of a circle is proportional to the square of its radius, r cm. That is, A r 2 or A kr 2 In general if is proportional to the square of 2 and k 2 where k is the constant of proportionalit. Since k 2, the graph is a quadratic curve passing through the origin. k is generall positive. Similarl if is proportional to the cube of 3 and k 3 where k is the constant of proportionalit. Eample 3 The area, A cm 2, of a square is proportional to the square of its perimeter, P cm. When P 8, A 4 Find a formula for A in terms of P. Solution 3 A P 2 so A kp 2 4 k k A P Substitute A 4, P 8 into A kp 2. This is the formula for A in terms of P. O k 2 Eample 4 is proportional to the square of. 60 when 6 a Find a formula for in terms of. b Find when 4.5 c Find a value of for which 135 Solution 4 a 2 so k 2 60 k 6 2 k Substitute 60, 6 into k 2 b Substitute 4.5 into c Substitute 135 into (or 9) 552

4 34.2 Further direct proportion CHAPTER 34 Eample 5 The mass, M kg, of a solid cube made from lead is proportional to the cube of the length, L cm, of an edge. When L 0.2, M 90 a Find a formula for M in terms of L. b Find the value of M when L 0.3 c Find the value of L when M 2000 Give our answer correct to 3 significant figures. Solution 5 a M L 3 so M kl 3 90 k k M L 3 b M M c L L L L (to 3 s.f.) Substitute M 90, L 0.2 into M kl 3 Substitute L 0.3 into M L 3 Substitute M 2000 into M L 3 Eercise 34B 1 is proportional to the square of. 2 is proportional to the square of. a When 2, 8 Find when 3 a When 2, 12 Find when 108 b When 2, 10 Find when 8 b When 3, 18 Find when 162 c When 2, 7 Find when 6 c When 4, 40 Find when 160 d When 3, 12 Find when 15 d When 4, 200 Find when 32 3 The area, A cm 2, of a regular heagon is 4 The rate of heat loss, H calories per second, proportional to the square of the length, from a sphere is proportional to the square l cm, of the longest diagonal. of the radius, r cm, of the sphere. When A 65, l 10 When H 2.5, r 5 a Find a formula for A in terms of l. a Find a formula for H in terms of r. b Find the value of A when l 4 b Find the value of H when r 4 c Find the value of l when A 200 c Find the value of r when H 90 Give our answer correct to 3 significant figures. 5 The quantit of light, Q,given out 6 The power, P watts, of an engine is b a lamp is proportional to the square proportional to the square of of the current, I, passing through the lamp. the speed, s m/s, of the engine. When Q 1000, I 2 When s 30, P 1260 a Find a formula for Q in terms of I. a Find a formula for P in terms of s. b Find the value of Q when I 3 b Find the value of P when s 25 c Find the value of I when Q 2000 c Find the value of s when P 1715 Give our answer correct to 3 significant figures. 553

5 CHAPTER 34 7 is proportional to the cube of. a When 2, 16 Find when 3 b When 2, 10 Find when 3 c When 4, 20 Find when 6 d When 5, 800 Find when 8 Direct and inverse proportion 8 is proportional to the cube of. When 8, 1000 a Find a formula for in terms of. b Find the value of when Inverse proportion When one quantit increases at the same rate as another quantit decreases, the quantities are said to be inversel proportional to each other (see Section 20.5). In general if is inversel proportional to 1 and k where k is the constant of proportionalit. The graph of k when k is positive has a similar shape to the graph of 1 (see Section 25.1). k Similarl if is inversel proportional to the square of 1 2 and k2 where k is the constant of proportionalit. O Here is the graph of when k is positive. k2 k 2 O Eample 6 When a fied volume of water is poured into a clindrical jar, the depth, D cm, of the water is inversel proportional to the cross-sectional area, A cm 2, of the clindrical jar. When A 40, D 120 a Find a formula for A in terms of D. b Find A when D 150 c Find D when A

6 34.3 Inverse proportion CHAPTER 34 Solution 6 a A D 1 so A D k k k A D b A c D D Substitute A 40, D 120 into A k D Substitute D 150 into A 4800 D Substitute A 60 into A 4800 D Eample 7 The force of attraction, F newtons, between two spheres is inversel proportional to the square of the distance, d m, between the centres of the spheres. When d 2, F a Epress F in terms of d. b Find F when d 2.5 c Find d when F Give our answer correct to 3 significant figures. Solution 7 a F d 1 2 so F d k2 k k F d b F c d d d d 4.90 (to 3 s.f.) Substitute F 0.006, d 2 into F k d 2 Substitute d 2.5 into F d 2 Substitute F into F d 2 Eercise 34C 1 is inversel proportional to. a 8 when 2 Find when 4 b 10 when 4 Find when 16 c 16 when 10 Find when 8 d 21 when 10 Find when

7 CHAPTER 34 Direct and inverse proportion 2 is inversel proportional to. a 20 when 4 Find when 5 b 25 when 8 Find when 10 c 30 when 9 Find when 20 d 45 when 4 Find when 54 3 The time, T seconds, taken for a pan of water to boil on a gas ring is inversel proportional to the setting, N, of the gas ring. When N 4.5, T 108 a Find a formula for T in terms of N. b Find the value of T when N 6 c If the pan takes 3 minutes to boil, what was the setting? 4 For rectangles with the same area, the length, l metres, of the rectangle is inversel proportional to the width, w metres, of the rectangle. When l 2.5, w 2.4 a Epress l in terms of w. b Find the value of w when l 3.2 c Given that the values of l and w are the same, find the value of l. 5 The frequenc, f ccles per second, of a sound wave is inversel proportional to the wavelength, l cm, of the sound wave. When f 256, l 133 a Find a formula for f in terms of l. b Find f when l 250 Give our answer correct to 3 significant figures. c Find l when f 300 Give our answer correct to 3 significant figures. 6 is inversel proportional to the square of. a 4 when 2 Find when 4 b 10 when 2 Find when 4 c 20 when 3 Find when 2 d 45 when 4 Find when 5 7 is inversel proportional to the square of. a 12 when 2 Find when 0.75 b 10 when 4 Find when 6.4 c 0.5 when 6 Find when 1800 d 12.5 when 2 Find when 2 8 When a fied volume of liquid is poured into an clinder, the depth, D cm, of the liquid is inversel proportional to the square of the radius, r cm, of the clinder. When r 5, D 40 a Find a formula for D in terms of r. b Find the value of D when r 4 c Find the value of r when D 15 Give our answer correct to 3 significant figures. d For what value of r is the depth equal to the diameter of the clinder? Give our answer correct to 3 significant figures. 9 The intensit, I, of the light at a distance, d,from a lamp is inversel proportional to the square of the distance. When I 4.5, d 2.4 a Find a formula for I in terms of d. b Find I when d 1.8 c Find d when I 6 Give our answer correct to 3 significant figures. 556

8 34.4 Proportion and square roots CHAPTER The pressure, P pascals, that a constant force eerts on a square with an edge of length, m, is inversel proportional to. When 0.4, P 50 a Find a formula for P in terms of. b Find P when 0.5 c Find when P 600 Give our answer correct to 3 significant figures Proportion and square roots Sometimes one quantit is directl proportional to the square root of another quantit. In general if is proportional to the square root of and k k where k is the constant of proportionalit. Here is the graph of k when k is positive. O Eample 8 The speed, s, of a particle is directl proportional to the square root of its kinetic energ, E. When E 225, s 40 a Find a formula for s in terms of E. b Find s when E 900 c Rearrange the formula to find E in terms of s. Solution 8 a s E so s k E 40 k 225 k 15 k s 8 3 E Substitute s 40, E 225 into s k E b s Substitute E 900 into s 8 E 3 c s 80 s 8 3 E E 3 s 8 E 3 2 s 8 or E 9 s2 64 Multipl both sides b 3 and then divide both sides b 8 Square both sides. Either formula is acceptable. Sometimes one quantit is inversel proportional to the square root of another quantit. In general if is inversel proportional to the square root of 1 k and where k is the constant of proportionalit. k Here is the graph of k when k is positive. O 557

9 CHAPTER 34 Direct and inverse proportion Eample 9 is inversel proportional to the square root of. When 64, 20 a Find a formula for in terms of. b Find when 100 c Find when 5 Solution 9 1 k a so k k k b Substitute 20, 64 into Substitute 100 into 160 k c , Substitute 5 into 160 Multipl both sides b Square both sides. Eercise 34D 1 is directl proportional to the square root of. a When 4, 6 Find when 25 b When 16, 20 Find when 49 c When 9, 4 Find when 81 d When 100, 40 Find when is directl proportional to the square root of. a When 1, 4 Find when 8 b When 4, 10 Find when 25 c When 16, 10 Find when 25 d When 49, 21 Find when 27 3 is inversel proportional to the square root of. a When 4, 2 Find when 25 b When 1, 5 Find when 16 c When 4, 4 Find when 1 d When 100, 0.3 Find when is inversel proportional to the square root of. a When 4, 2.5 Find when 2 b When 4, 1 2 Find when 2 c When 9, 2 Find when 6 d When 25, 0.8 Find when

10 Chapter 34 review questions CHAPTER 34 5 When a ball is thrown upwards, the time, T seconds, the ball remains in the air is directl proportional to the square root of the height, h metres, reached. When h 25, T 4.47 a Find a formula for T in terms of h. b Find the value of T when h 50 Give our answer correct to 3 significant figures. The ball is thrown upwards and remains in the air for 5 seconds. c Find the height reached. Give our answer correct to 3 significant figures. Chapter summar You should now know: how to set up and use equations to solve problems involving direct proportion, for eample if is directl proportional to, and k if is directl proportional to the square of, 2 and k 2 how to set up and use equations to solve problems involving inverse proportion, for eample if is inversel proportional to, 1 and k if is inversel proportional to the square of, 1 2 and k2 that k is a number known as the constant of proportionalit the shapes of the graphs that represent the different tpes of proportionalit. Chapter 34 review questions 1 Here are three eamples of proportionalit. i is directl proportional to. ii V is directl proportional to the cube of r. iii T is inversel proportional to the square root of s. a Epress each of i to iii as a formula. Include a constant of proportionalit. b Draw a sketch of the graph that represents the tpe of proportionalit described in each of i to iii. 2 V is directl proportional to r. When r 2, V 8 a Find V when r 6 b Find r when V 2 3 is inversel proportional to. When 10, 12 a Find a formula for in terms of. b Find the value of when 20 c Find the value of when 25 4 The time, T seconds, it takes a pendulum to swing once is proportional to the square root of the length, l metres, of the pendulum. When l 0.16, T 0.8 a Find a formula for T in terms of l. b Find the value of T when l 1.44 c What length of pendulum will give a swing of 1 second? Give our answer correct to 3 significant figures. 559

11 CHAPTER 34 Direct and inverse proportion 5 The drag force, F newtons, on an object moving with a speed, s metres per second, is proportional to the square of the speed. When s 20, F 80 a Epress F in terms of s. b Find F when s 30 c Find the speed when the drag force is 300 newtons. 6 d is directl proportional to the square of t. d 80 when t 4 a Epress d in terms of t. b Work out the value of d when t 7 c Work out the positive value of t when d 45 (1387 June 2005) 7 P is inversel proportional to V. When V 2, P 7.5 a Find a formula for P in terms of V. The value of V is increased b 25%. b Work out the percentage change in the value of P. 8 The temperature, T, at a distance, d metres, from a heat source is inversel proportional to the square of the distance. When d 4, T 275 a Find T when d 6 b Find d when T 1000 Give our answer correct to 3 significant figures. 9 The oscillation frequenc, f ccles per second, of a spring is inversel proportional to the square root of the mass, m kg, of the spring. When m 2.56, f 2 Find f when m 4 10 The time taken, T seconds, for a particle to slide down a smooth slope of length, l m, is directl proportional to the square root of the length. When T 1.5, l 6.25 a Find a formula for T in terms of l. b Rearrange the formula to find l in terms of T. 11 The rate of melting, M grams per second, of a sphere of ice is proportional to the square of the radius, r cm. When r 20, M 0.6 a Show that M r 2 b Find the rate of melting when the radius is 40 cm. c Find the radius when the rate of melting is 1 gram per second. Give our answer correct to 3 significant figures. d Hannah claims that the rate of melting is directl proportional to the surface area, A cm 2,of the sphere. Is Hannah correct? You must justif our answer. 12 The force, F, between two magnets is inversel proportional to the square of the distance,, between them. When 3, F 4 a Find an epression for F in terms of. b Calculate F when 2 c Calculate when F 64 (1387 June 2003) 560

12 Chapter 34 review questions CHAPTER In a factor, chemical reactions are carried out in spherical containers. The time, T minutes, the chemical reaction takes is directl proportional to the square of the radius, R cm, of the spherical container. When R 120, T 32 Find the value of T when R 150 (1387 November 2004) 14 The shutter speed, S, of a camera varies inversel as the square of the aperture setting, f. When f 8, S 125 a Find a formula for S in terms of f. b Hence, or otherwise, calculate the value of S when f 4 (1387 June 2004) 15 Graph A Graph B Graph C Graph D The graphs of against represent four different tpes of proportionalit. Cop the table and write down the letter of the graph which represents the tpe of proportionalit. Tpe of proportionalit Graph letter is directl proportional to... is inversel proportional to... is proportional to the square of... is inversel proportional to the square of... (1387 November 2004) 561