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1 1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion Wht you should lredy know Squres, squre roots, cues nd cue roots of integers How to sustitute vlues into lgeric expressions How to solve simple lgeric equtions Quick check 1 Write down the vlue of ech of the following. 5 2 «««81 c 3 3 d 3 «««64 2 Clculte the vlue of y if x = 4. y = 3x 2 y = 1 «x 519

2 22.1 Direct vrition This section will introduce you to: direct vrition nd show you how to work out the constnt of proportionlity Key words constnt of proportionlity, k direct proportion direct vrition The term direct vrition hs the sme mening s direct proportion. There is direct vrition (or direct proportion) etween two vriles when one vrile is simple multiple of the other. Tht is, their rtio is constnt. For exmple: 1 kilogrm = 2.2 pounds There is multiplying fctor of 2.2 etween kilogrms nd pounds. Are of circle = πr 2 There is multiplying fctor of π etween the re of circle nd the squre of its rdius. An exmintion question involving direct vrition usully requires you first to find this multiplying fctor (clled the constnt of proportionlity), then to use it to solve prolem. The symol for vrition or proportion is. So the sttement Py is directly proportionl to time cn e mthemticlly written s Py Time which implies tht Py = k Time where k is the constnt of proportionlity. There re three steps to e followed when solving question involving proportionlity. Step 1: set up the proportionlity eqution (you my hve to define vriles). Step 2: use the given informtion to find the constnt of proportionlity. Step 3: sustitute the constnt of proportionlity in the originl eqution nd use this to find unknown vlues. 520

3 CHAPTER 22: VARIATION EXAMPLE 1 The cost of n rticle is directly proportionl to the time spent mking it. An rticle tking 6 hours to mke costs 30. Find the following. the cost of n rticle tht tkes 5 hours to mke the length of time it tkes to mke n rticle costing 40 Step 1: Let C e the cost of mking n rticle nd t the time it tkes. We then hve: C t C = kt where k is the constnt of proportionlity. Note tht we cn replce the proportionlity sign with = k to otin the proportionlity eqution. Step 2: Since C = 30 when t = 6 hours, then 30 = 6k 30 = k 6 k = 5 Step 3: So the formul is C = 5t When t = 5 hours C = 5 5 = 25 So the cost is 25. When C = = 5 t 40 5 = t t = 8 So the mking time is 8 hours. EXERCISE 22A In ech cse, first find k, the constnt of proportionlity, nd then the formul connecting the vriles. T is directly proportionl to M. If T = 20 when M = 4, find the following. T when M = 3 M when T = 10 W is directly proportionl to F. If W = 45 when F = 3, find the following. W when F = 5 F when W = 90 Q vries directly with P. If Q = 100 when P = 2, find the following. Q when P = 3 P when Q = 300 X vries directly with Y. If X = 17.5 when Y = 7, find the following. X when Y = 9 Y when X =

4 CHAPTER 22: VARIATION The distnce covered y trin is directly proportionl to the time tken. The trin trvels 105 miles in 3 hours. Wht distnce will the trin cover in 5 hours? Wht time will it tke for the trin to cover 280 miles? The cost of fuel delivered to your door is directly proportionl to the weight received. When 250 kg is delivered, it costs How much will it cost to hve 350 kg delivered? How much would e delivered if the cost were 33.25? The numer of children who cn ply sfely in plyground is directly proportionl to the re of the plyground. A plyground with n re of 210 m 2 is sfe for 60 children. How mny children cn sfely ply in plyground of re 154 m 2? A plygroup hs 24 children. Wht is the smllest plyground re in which they could sfely ply? Direct proportions involving squres, cues nd squre roots The process is the sme s for liner direct vrition, s the next exmple shows. EXAMPLE 2 The cost of circulr dge is directly proportionl to the squre of its rdius. The cost of dge with rdius of 2 cm is 68p. Find the cost of dge of rdius 2.4 cm. Find the rdius of dge costing Step 1: Let C e the cost nd r the rdius of dge. Then C r 2 C = kr 2 where k is the constnt of proportionlity. Step 2: C = 68p when r = 2 cm. So 68 = 4k 68 = k k = 17 4 Hence the formul is C = 17r 2 When r = 2.4 cm C = p = 97.92p Rounding off gives the cost s 98p. When C = 153p 153 = 17r = 9 = r 2 17 r = ««9 = 3 Hence, the rdius is 3 cm. 522

5 CHAPTER 22: VARIATION EXERCISE 22B In ech cse, first find k, the constnt of proportionlity, nd then the formul connecting the vriles. T is directly proportionl to x 2. If T = 36 when x = 3, find the following. T when x = 5 x when T = 400 W is directly proportionl to M 2. If W = 12 when M = 2, find the following. W when M = 3 M when W = 75 E vries directly with ««C. If E = 40 when C = 25, find the following. E when C = 49 C when E = 10.4 X is directly proportionl to ««Y. If X = 128 when Y = 16, find the following. X when Y = 36 Y when X = 48 P is directly proportionl to f 3. If P = 400 when f = 10, find the following. P when f = 4 f when P = 50 The cost of serving te nd iscuits vries directly with the squre root of the numer of people t the uffet. It costs 25 to serve te nd iscuits to 100 people. How much will it cost to serve te nd iscuits to 400 people? For cost of 37.50, how mny could e served te nd iscuits? In n experiment, the temperture, in C, vried directly with the squre of the pressure, in tmospheres. The temperture ws 20 C when the pressure ws 5 tm. Wht will the temperture e t 2 tm? Wht will the pressure e t 80 C? The weight, in grms, of ll erings vries directly with the cue of the rdius mesured in millimetres. A ll ering of rdius 4 mm hs weight of g. Wht will ll ering of rdius 6 mm weigh? A ll ering hs weight of 48.6 g. Wht is its rdius? The energy, in J, of prticle vries directly with the squre of its speed in m/s. A prticle moving t 20 m/s hs 50 J of energy. How much energy hs prticle moving t 4 m/s? At wht speed is prticle moving if it hs 200 J of energy? The cost, in, of trip vries directly with the squre root of the numer of miles trvelled. The cost of 100-mile trip is 35. Wht is the cost of 500-mile trip (to the nerest 1)? Wht is the distnce of trip costing 70? 523

6 22.2 Inverse vrition This section will introduce you to: inverse vrition nd show you how to work out the constnt of proportionlity Key words constnt of proportionlity, k inverse proportion There is inverse vrition etween two vriles when one vrile is directly proportionl to the reciprocl of the other. Tht is, the product of the two vriles is constnt. So, s one vrile increses, the other decreses. For exmple, the fster you trvel over given distnce, the less time it tkes. So there is n inverse vrition etween speed nd time. We sy speed is inversely proportionl to time. 1 k S nd so S = T T which cn e written s ST = k. EXAMPLE 3 M is inversely proportionl to R. If M = 9 when R = 4, find the following. M when R = 2 R when M = 3 1 k Step 1: M M = where k is the constnt of proportionlity. R R k Step 2: When M = 9 nd R = 4, we get 9 = = k k = Step 3: So the formul is M = R 36 When R = 2, then M = = When M = 3, then 3 = 3R = 36 R = 12 R EXERCISE 22C In ech cse, first find the formul connecting the vriles. T is inversely proportionl to m. If T = 6 when m = 2, find the following. T when m = 4 m when T = 4.8 W is inversely proportionl to x. If W = 5 when x = 12, find the following. W when x = 3 x when W =

7 CHAPTER 22: VARIATION Q vries inversely with (5 t). If Q = 8 when t = 3, find the following. Q when t = 10 t when Q = 16 M vries inversely with t 2. If M = 9 when t = 2, find the following. M when t = 3 t when M = 1.44 W is inversely proportionl to ««T. If W = 6 when T = 16, find the following. W when T = 25 T when W = 2.4 The grnt ville to section of society ws inversely proportionl to the numer of people needing the grnt. When 30 people needed grnt, they received 60 ech. Wht would the grnt hve een if 120 people hd needed one? If the grnt hd een 50 ech, how mny people would hve received it? While doing underwter tests in one prt of n ocen, tem of scientists noticed tht the temperture in C ws inversely proportionl to the depth in kilometres. When the temperture ws 6 C, the scientists were t depth of 4 km. Wht would the temperture hve een t depth of 8 km? To wht depth would they hve hd to go to find the temperture t 2 C? A new engine ws eing tested, ut it hd serious prolems. The distnce it went, in km, without reking down ws inversely proportionl to the squre of its speed in m/s. When the speed ws 12 m/s, the engine lsted 3 km. Find the distnce covered efore rekdown, when the speed is 15 m/s. On one test, the engine roke down fter 6.75 km. Wht ws the speed? In lloon it ws noticed tht the pressure, in tmospheres, ws inversely proportionl to the squre root of the height, in metres. When the lloon ws t height of 25 m, the pressure ws 1.44 tm. Wht ws the pressure t height of 9 m? Wht would the height hve een if the pressure ws 0.72 tm? The mount of wste which firm produces, mesured in tonnes per hour, is inversely proportionl to the squre root of the size of the filter eds, mesured in m 2. At the moment, the firm produces 1.25 tonnes per hour of wste, with filter eds of size 0.16 m 2. The filter eds used to e only 0.01 m 2. How much wste did the firm produce then? How much wste could e produced if the filter eds were 0.75 m 2? 525

8 y is proportionl to x. Complete the tle. x y The energy, E, of n oject moving horizontlly is directly proportionl to the speed, v, of the oject. When the speed is 10 m/s the energy is Joules. Find n eqution connecting E nd v. Find the speed of the oject when the energy is Joules. y is inversely proportionl to the cue root of x. When y = 8, x = 1 8. Find n expression for y in terms of x, Clculte i the vlue of y when x = 1 125, ii the vlue of x when y = 2. The mss of cue is directly proportionl to the cue of its side. A cue with side of 4 cm hs mss of 320 grms. Clculte the side length of cue mde of the sme mteril with mss of grms y is directly proportionl to the cue of x. When y = 16, x = 3. Find the vlue of y when x = 6. d is directly proportionl to the squre of t. d = 80 when t = 4 Express d in terms of t. Work out the vlue of d when t = 7. c Work out the positive vlue of t when d = 45. Edexcel, Question 16, Pper 5 Higher, June 2005 The force, F, etween two mgnets is inversely proportionl to the squre of the distnce, x, etween them. When x = 3, F = 4. Find n expression for F in terms of x. Clculte F when x = 2. c Clculte x when F = 64. Edexcel, Question 17, Pper 5 Higher, June Two vriles, x nd y, re known to e proportionl to ech other. When x = 10, y = 25. Find the constnt of proportionlity, k, if: y x y x 2 c y 1 x d ««y 1 x y is directly proportionl to the cue root of x. When x = 27, y = 6. Find the vlue of y when x = 125. Find the vlue of x when y = 3. The surfce re, A, of solid is directly proportionl to the squre of the depth, d. When d = 6, A = 12π. Find the vlue of A when d = 12. Give your nswer in terms of π. Find the vlue of d when A = 27π. r is inversely proportionl to t. r = 12 when t = 0.2 Clculte the vlue of r when t = 4. Edexcel, Question 4, Pper 13B Higher, Jnury 2003 The frequency, f, of sound is inversely proportionl to the wvelength, w. A sound with frequency of 36 hertz hs wvelength of metres. Clculte the frequency when the frequency nd the wvelength hve the sme numericl vlue. t is proportionl to m 3. When m = 6, t = 324. Find the vlue of t when m = 10. Also, m is inversely proportionl to the squre root of w. When t = 12, w = 25. Find the vlue of w when m = 4. P nd Q re positive quntities. P is inversely proportionl to Q 2. When P = 160, Q = 20. Find the vlue of P when P = Q. 526

9 CHAPTER 22: VARIATION WORKED EXAM QUESTION y is inversely proportionl to the squre of x. When y is 40, x = 5. Find n eqution connecting x nd y. Find the vlue of y when x = 10. Solution 1 y y = x 2 k x 2 k 40 = 25 k = = y = x 2 or yx 2 = When x = 10, y = = = First set up the proportionlity reltionship nd replce the proportionlity sign with = k. Sustitute the given vlues of y nd x into the proportionlity eqution to find the vlue of k. Sustitute the vlue of k to get the finl eqution connecting y nd x. Sustitute the vlue of x into the eqution to find y. The mss of solid, M, is directly proportionl to the cue of its height, h. When h = 10, M = The surfce re, A, of the solid is directly proportionl to the squre of the height, h. When h = 10, A = 50. Find A, when M = Solution M = kh = k 1000 k = 4 So, M = 4h 3 A = ph 2 50 = p 100 p = 1 So, A = h = 4h 3 h 3 = 8000 h = A = (20) = = First, find the reltionship etween M nd h using the given informtion. Next, find the reltionship etween A nd h using the given informtion. Find the vlue of h when M = Now find the vlue of A for tht vlue of h. 527

10 An electricity compny wnts to uild some offshore wind turines (s shown elow). The compny is concerned out how ig the turines will look to person stnding on the shore. It sks n engineer to clculte the ngle of elevtion from the shore to the highest point of turine, when it is rotting, if the turine ws plced t different distnces out to se. Help the engineer to complete the first tle elow. Distnce of Angle of elevtion turine out to se from shore 3km 2.29º 4km 5km 6km 7km 8km 50m 70m The power ville in the wind is mesured in wtts per metre squred of rotor re (W/m 2 ). Wind speed is mesured in metres per second (m/s). The power ville in the wind is proportionl to the cue of its speed. A wind speed of 7 m/s cn provide 210 W/m 2 of energy. Complete the tle elow to show the ville power t different wind speeds. Wind speed Aville power (m/s) (W/m 2 )

11 Vrition The engineer investigtes the different mounts of power produced y different length rotor ldes t different wind speeds. He clcultes the rotor re for ech lde length this is the re of the circle mde y the rotors nd then works out the power produced y these ldes t the different wind speeds shown. Help him to complete the tle. Rememer 1 W = 1 wtt 1000 W = 1 kw = 1 kilowtt 1000 kw = 1 MW = 1 megwtt Wind speed Aville power Rotor re for Power Rotor re for Power Rotor re for Power (m/s) (W/m 2 ) 50 m lde (m 2 ) (MW) 60 m lde (m 2 ) (MW) 70 m lde (m 2 ) (MW)

12 GRADE YOURSELF Ale to find formule descriing direct or inverse vrition nd use them to solve prolems Ale to solve direct nd inverse vrition prolems involving three vriles Wht you should know now How to recognise direct nd inverse vrition Wht constnt of proportionlity is, nd how to find it How to find formule descriing inverse or direct vrition How to solve prolems involving direct or inverse vrition 530

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