Mthemtis SKE: STRAND F UNIT F1 Formule: Tet STRAND F: Alger F1 Formule Tet Contents Setion F1.1 Using Formule F1. Construt nd Use Simple Formule F1.3 Revision of Negtive Numers F1.4 Sustitution into Formule F1.5 More Comple Formule F1.6 Chnging the Sujet F1.7 More Chnge of Sujet CIMT, Plmouth Universit
Mthemtis SKE, Strnd F UNIT F1 Formule: Tet F1 Formule F1.1 Using Formule In formule, letters re used to represent numers. For emple, the formul A lw l n e used to find the re of retngle. Here A is the re, l the length nd w the width. In this formul, lw mens l w. w Formule re usull written in this w, without multiplition signs. The perimeter of the retngle would e given the formul P l + w Here gin there re no multiplition signs, nd l mens l nd w mens w. Worked Emple 1 The perimeter of retngle n e found using the formul Find the perimeter if l P l + w 8 nd w 4. The letters l nd w should e repled the numers 8 nd 4. This gives P 8 + 4 16 + 8 Worked Emple 4 The finl speed of r is v nd n e lulted using the formul v u + t where u is the initil speed, is the elertion nd t is the time tken. Find v if the elertion is m s, the time tken is 10 seonds nd the initil speed is 4 m s 1. The elertion is m s so. The initil speed is 4 m s 1 so u 4. The time tken is 10 s so t 10. Using the formul v u + t gives CIMT, Plmouth Universit 1 v 4 + 10 4 + 0 4 m s 1
F1.1 Mthemtis SKE, Strnd F UNIT F1 Formule: Tet Eerises 1. The re of retngle is found using the formul A lw nd the perimeter using P l + w. Find the re nd perimeter if: () l 4 nd w () l 10 nd w 3 () l 11 nd w (d) l 5 nd w 4. The formul v u + t is used to find the finl speed. Find v if`: () u 6, nd t 5 () u 0, 4 nd t 3 () u 3, 1 nd t 1 (d) u 1, nd t 4 3. Use the formul F m to find F if: () m 10 nd 3 () m 00 nd 4. The perimeter of tringle is found using the formul Find P if: P + + () 10, 1 nd 8 () 3, 4 nd 5 () 6, 4 nd 7 5. The volume of o is given the formul V Find V if: (), 3 nd 10 () 7, 5 nd 3 () 4, 4 nd 9 6. Find the vlue of Q for eh formul using the vlues given. () Q 3 + 7 () Q + 4 nd 3 nd 5 () Q + 4 (d) Q 5 3 nd 5 10 nd CIMT, Plmouth Universit
F1.1 Mthemtis SKE, Strnd F UNIT F1 Formule: Tet (e) Q (f) Q 10 nd 4 nd (g) Q + 4 (h) Q 4 + 8 nd 3 5 nd 11 (i) Q 3 + + z (j) Q + z 4, nd z 10, 5 nd z 8 (k) Q z (l) Q + 4 z, 5 nd z 3 8, 3 nd z 4 (m) Q + z (n) Q + z 8, 10 nd z 3 50, nd z 3 7. This formul is used to work out Alele's p: P 0 + Numer of hours worked Rte of p Her rte of p is 0 plus 9 per hour. Work out her p for 40 hours. 8. A retngle hs length of m nd width of m. ( ) The perimeter of retngle is given the formul p +. Clulte the perimeter of retngle when 45. nd 4.. F1. Construt nd Use Simple Formule A formul desries how one quntit reltes to one or more other quntities. For emple, formul for the re of retngle desries how to find the re, given the length nd width of the retngle. The perimeter of the retngle would e given the formul P l + w Here gin there re no multiplition signs nd l mens l nd w mens w. Worked Emple 1 () Write down formul for the perimeter of the shpe shown. () Find the perimeter if m, 3 m nd 5 m CIMT, Plmouth Universit 3
F1. Mthemtis SKE, Strnd F UNIT F1 Formule: Tet () The perimeter is found dding together the lengths of ll the sides, so the formul will e P + + + + ut s nd re oth dded in twie, this n e simplified to P + + () If m, 3 m nd 5 m, Worked Emple P + 3 + 5 4 + 6 + 5 15 m When ling ptio, lndspe grdener hrges si fee of 30 plus 1 per hour. Find formul for lulting the grdener's hrge. Let C hrge nd n numer of hours. The hrge is mde up of fied 30 nd 1 the numer of hours or 1n. So the totl hrge in is given C 30 + 1 n Eerises 1. Find formul for the perimeter of eh shpe, nd find the perimeter for the speified vlues. () () 6 m, 4 m 5 m () (d) 6 m, 10 m 5 m, 6 m, 10 m CIMT, Plmouth Universit 4
F1. Mthemtis SKE, Strnd F UNIT F1 Formule: Tet (e) (f) 10 m 4 m, 5 m, 9 m (g) (h) 60 m, 160 m, 4 m, 9 m 80 m. Find formul for the re of eh of the shpes elow nd find the re for the vlues given. () () 6 m, 10 m 3 m () (d) m, 8 m 3 m, 4 m, 9 m (e) (f) 4 m, 5 m CIMT, Plmouth Universit 5 50 m, 00 m
F1. Mthemtis SKE, Strnd F UNIT F1 Formule: Tet 3. Three onseutive numers re to e dded together. () If is the smllest numer, wht re the other two numers in terms of? () Write down formul for the totl, T, of the three numers in terms of, using our nswer to (). 4. () Write down formul to find the men, M, of the two numers nd. () Write down formul to find the men, M, of the five numers p, q, r, s, nd t. 5. Tikets for shool onert re sold t 6 for dults nd 4 for hildren. () If p dults nd q hildren u tikets, write formul for the totl vlue, T, of the tiket sles. () Find the totl vlue of the tiket sles if p 50 nd q 0. 6. A retngle is 3 m longer thn it is wide. If is the width, write down formul for: () the perimeter, P; () the re, A, of the retngle. 7. Rhel is one er older thn Brdle. Crl is three ers ounger thn Brdle If Brdle is ers old, write down epressions for: () Rhel's ge; () Crl's ge; () the sum of ll three hildren's ges. 8. A window lener hrges fee of 8 for visiting house nd 4 for ever window tht he lens. () Write down formul for finding the totl ost C, in, when n windows re lened. () Find C if n 8. 9. A ti driver hrges fee of 5, plus 4 for ever km tht the ti trvels. () Find formul for the ost C of journe tht overs km. () Find C if 3. CIMT, Plmouth Universit 6
F1. Mthemtis SKE, Strnd F UNIT F1 Formule: Tet 10. A grdener uilds pths using pving sls lid out in pttern s shown, with white sls on eh side of row of red sls. () () If n red sls re used, how mn white sls re needed? Another grdener puts white sl t eh end of the pth s shown elow. If n red sls re used, how mn white sls re needed? 11. A pth of width is lid round retngulr lwn s shown. 0 () () Find n epression for the perimeter of the grss. Find n epression for the re of the grss. grss 30 1. Juie drinks ost 7 p eh. Write down formul for the ost, C p, of n drinks. pth 13. () Fuel osts 80 p per litre. Write down formul for the ost, C p, of l litres of fuel. () Fuel osts p per litre. Write down formul for the ost, C p, of l litres of fuel. 14. Write down n epression for the TOTAL ost, in pounds, of 8 metres of fri t pounds per metre nd reels of thred t pounds per reel. 15. Mr Jmes works si week of 40 hours t rte of 16 n hour. His overtime rte is 4 per hour MORE thn his si rte. Clulte (i) his TOTAL wge for si week. (ii) his wge for week in whih he worked 47 hours. F1.3 Revision of Negtive Numers Before strting the net setion on formule it is useful to revise how to work with negtive numers. Note When multipling or dividing two numers, if the hve the sme sign the result will e positive, ut if the hve different signs the result will e negtive. CIMT, Plmouth Universit 7
F1.3 Mthemtis SKE, Strnd F UNIT F1 Formule: Tet Worked Emple 1 Find () () ( 3) ( 7 ) () ( 4) 3 ( 40) ( 5 ) (d) ( 6) 7 () () ( 3) ( 7) 1 () ( 4) 3 8 ( 40) ( 5) 8 (d) ( 6) 7 4 Note When dding or sutrting it n e helpful to use numer line, rememering to move up when dding nd down when sutrting positive numer. When dding negtive numer, move down nd when sutrting negtive numer, move up. Worked Emple Find () 4 10 () 6 + 8 ( ) () 4 5 (d) 6+ 7 (e) 7 ( 4) () 4 10 6 () 6 + 8 + 4 3 1 0 1 3 4 Numer line ( ) () 4 5 9 (d) 6+ 7 6 7 (e) 7 ( 4) 7 + 4 Eerises 11 13 1. () 6 8 () 8 + 1 () 5 + ( ) ( ) ( ) ( ) (d) 6 (e) 8 3 (f) 9 6 (g) ( 4) ( 3 ) (h) 16 ( ) (i) ( 81) ( 3) (j) 16 + 4 (k) 8 5 (l) ( 5) 7 (m) 3 ( 8) (n) 1 10 (o) 10 + 5 (p) 9 + 6 (s) + ( ) ( ) (q) 4 ( 7) (r) 1 ( 4) ( ) ( ) 1 7 (t) 4 + (u) 6 5 CIMT, Plmouth Universit 8
F1.3 Mthemtis SKE, Strnd F UNIT F1 Formule: Tet. () ( ) ( ) ( ) + ( ) 1 () 4 () 4 3 3. In London, the temperture t midd ws 5 C. At midnight the temperture hd fllen 8 C. Wht ws the temperture t midnight? 4. The temperture ws reorded inside nd outside house in New York. Inside temperture Outside temperture 16 C 8 C How mn degrees wrmer ws it inside the house thn outside? Chllenge! You open ook. Two pges fe ou. If the produt of the two pge numers is 319, wht re the two pge numers? F1.4 Sustitution into Formule The proess of repling the letters in formul is known s sustitution. Worked Emple 1 The length of metl rod is l m. The length hnges with temperture nd n e found the formul l 40 + 0. 0 T where T is the temperture. Find the length of the rod when () T 50 C nd () T 10 C () Using T 50 gives () Using T 10 gives l 40 + 50 0. 0 40 + 1 41 m l 40 + ( 10) 0. 0 ( ) 40 + 0. 40 0. 39. 8 m CIMT, Plmouth Universit 9
F1.4 Mthemtis SKE, Strnd F UNIT F1 Formule: Tet Worked Emple The profit in mde slesmn when he sells n ooks is lulted the formul P 4n 50 Find the profit if he mkes () 30 sles () 9 sles () Here n 30 so the formul gives () P 4 30 50 10 50 70 So the slesmn's profit fter 30 sles is 70. Here n 9 so the formul gives P 4 9 50 36 50 14 He mkes loss of 14 if onl 9 sles re mde. Worked Emple 3 If * is defined * 4 wht is the vlue of * when () 4, (), 4? () 4 * ( ) 4 4( ) 16 4 4 16 16 0 () ( ) 4 ( ) * 4 4 4 4 16 4 64 60 CIMT, Plmouth Universit 10
F1.4 Mthemtis SKE, Strnd F UNIT F1 Formule: Tet Eerises 1. The formul elow is used to onvert tempertures in degrees Celsius to degrees Fhrenheit, where F is the temperture in degrees Fhrenheit nd C is the temperture in degrees Celsius. Find F if: F 18. C + 3 () C 10 () C 0 () C 10 (d) C 5 (e) C 0 (f) C 15. The formul 1 s ( u + v) t is used to lulte the distne, s, tht n ojet trvels if it strts with veloit u nd hs veloit v, t seonds lter. Find s if: () u, v 8, t () u 3, v 5, t 10 () u 1., v 38., t 45. (d) u 4, v 8, t (e) u 4, v 8, t 5 (f) u 16., v 8., t 3. 3. The length, l, of spring is given the formul l 0 0. 08 F where F is the size of the fore pplied to the spring to ompress it. Find l if: () F 5 () F 0 () F 4 (d) F 15 4. The formul P 10n 400 gives the profit, P in, mde when n phones re sold in d t shop. Find P if: () n 1 () n 3 () n 4 (d) n 10 How mn phones must e sold to mke profit? 5. Work out the vlue of eh funtion sustituting the vlues given, without using lultor. () V p + q p 8 q 4 () p 10 nd 7 nd CIMT, Plmouth Universit 11
F1.4 Mthemtis SKE, Strnd F UNIT F1 Formule: Tet () z + (d) Q 10 nd 6 15 nd 6 (e) (g) (i) (k) P + (f) Q 4 nd 10 100 nd 4 + + z 1 1 V (h) R + 5, 5 nd z 8 4 nd S + (j) R 0. + 04. 3, 4 nd 16 10 nd 0 T + 5 (l) C + 0 nd 40 10 nd 5 (m) (o) P (n) A 10 nd 4, 3 nd 100 + X (p) z + 10, 17. nd 1. 3 nd 4 (q) P (r) Q + + z 10 nd 6 10, 5 nd z 10 6. Work out the vlue of eh funtion sustituting the vlues given, using lultor if neessr. () () P () V z + 10,. 0 49. nd 31. nd z 1. R (d) D + 4 36. nd 16. 04. nd 08. + (e) Q 3 + (f) V 5 + 37. nd 59. 16. nd 4. CIMT, Plmouth Universit 1
F1.4 Mthemtis SKE, Strnd F UNIT F1 Formule: Tet 7. If * 4, wht is the vlue of * when (), () 3, 1 8. The formul to onvert tempertures from degrees Fhrenheit ( F ) into degrees Celsius ( C) is 5 C ( F 3) 9 Clulte the temperture in degrees Celsius whih is equivlent to temperture of 4 F. 9. Given tht, 3 nd 0, evlute (i) 4 + 3 (ii) 1 3 10. Given tht m, p, t, lulte 4 ( m + p) () mp+ t () t Chllenge! There re 10 nk notes ltogether. The onsist of 10, 0 nd 50 notes. If the totl vlue of the notes is 180, find the numer of eh tpe of notes. F1.5 More Comple Formule Some formule suh s 1 1 1 + nd z + f u v rise in siene or mthemtis, ut when used do not led diretl to vlues of f or z. Here we show how to use the formul to lulte these vlues. Worked Emple 1 Use the formul to find f if u 10 nd v 8. 1 1 1 + f u v CIMT, Plmouth Universit 13
F1.5 Mthemtis SKE, Strnd F UNIT F1 Formule: Tet Sustituting into the formul gives 1 1 + f 10 First dd together the two frtions using 40 s ommon denomintor: 1 4 + f 40 1 9 f 40 1 8 5 40 Now to find f, turn oth frtions upside-down to give Worked Emple Find z using the formul if 36. nd 48.. f 1 40 or f 4 4 9 9 z + Sustituting these vlues into the formul gives z 36. + 48. z 1. 96 + 3. 04 z 36 Now the squre root n e tken of oth sides to give Eerises 1. Use the formul to find f if: z + 36 or 36 z 6 or 6 1 1 1 + f v u () v 3 nd u 4 () v 6 nd u 5 () v 7 nd u 3 (d) v 10 nd u 4 CIMT, Plmouth Universit 14
F1.5 Mthemtis SKE, Strnd F UNIT F1 Formule: Tet. Find z using the formul if: z + () 1. nd 05. () 48. nd 64. () 3 nd 1. 6 3. Find the vlue of z s frtion or mied numer in eh se elow. () () (e) 1 z + () 1 z + 4 nd 10 3 nd 4 1 3 + (d) z 4 5 1 z nd 7 nd 3 1 3 1 1 + (f) z 4 z + 1 5 nd + (g) 1 z + 4 (h) + 1 z 1 4 4 nd 1 (i) 3 1 + z 1 nd 6 4. Find z in eh se elow. () z 9 + () z + 4 147 nd 3 () z (d) z 44 nd 5 363 nd 3 (e) z + 6 (f) z 8 + 6 nd 3 16. 9 nd 7. 9 CIMT, Plmouth Universit 15
F1.5 Mthemtis SKE, Strnd F UNIT F1 Formule: Tet 5. When three resistors re onneted in prllel the totl resistne R is given 1 1 1 1 + + R X Y Z where X, Y nd Z re the resistnes of eh resistor. Find R if: () X 10, Y 0 nd Z 30 () X 1000, Y 5000 nd Z 000 () X 1500, Y 00 nd Z 1600 6. Use the formul 1 ( t v ) to lulte the vlue of given tht 50, t 5. nd v 06. Give our nswer orret to 1 deiml ple. Show ll neessr working. 7. The formul f uv u + v is used in the stud of light. () Clulte f when u 14. 9 nd v 10.. Give our nswer orret to 3 signifint figures. () B rounding the vlues of u nd v in prt () to signifint figures, hek whether our nswer to prt () is resonle. Show our working. Investigtion 3 3 3 3 Find four integers,,, nd d suh tht + + d. F1.6 Chnging the Sujet Sometimes formul n e rerrnged into more useful formt. For emple, the formul F 18. C + 3 n e used to onvert tempertures in degrees Celsius to degrees Fhrenheit. It n e rerrnged into the form C... to enle tempertures in degrees Fhrenheit to e onverted to degrees Celsius. We s tht the formul hs een rerrnged to mke C the sujet of the formul. CIMT, Plmouth Universit 16
F1.6 Mthemtis SKE, Strnd F UNIT F1 Formule: Tet Worked Emple 1 Rerrnge the formul to mke C the sujet of the formul. F 18. C + 3 The im is to remove ll terms from the right hnd side of the eqution eept for the C. First sutrt 3 from oth sides, whih gives Then dividing oth sides 1.8 gives So the formul n e rerrnged s Worked Emple F 3 1. 8C F 3 18. C F 3 C 18. The distne, s, trvelled r in time t from initil speed u to finl speed v is given the formul ( u + v) t s Mke v the sujet of the formul. First multipl oth sides of the formul to give Then divide oth sides t, to give Finll, sutrt u from oth sides to give So the formul eomes s ( u + v) t s t s t u + v u v s v u t CIMT, Plmouth Universit 17
F1.6 Eerises Mthemtis SKE, Strnd F UNIT F1 Formule: Tet 1. Mke the sujet of eh of the following formule. () 4 () + 3 () 4 8 (d) + 4 (e) 5 (f) + (g) (h) + (i) + (j) (k) + + (l) + (m) (n) + (o) 4 3 (p) (s) p d ( ) 3 4 q (q) ( + ) (r) (t) v 5( + ) 4 (u) z + ( ) 3 + 4 ( 3) 4. Ohm's lw is used in eletril iruits nd sttes tht V IR Write formule with I nd R s their sujets. 3. Newton's Seond lw sttes tht F m. Write formule with m nd s their sujets. 4. The formul C π r n e used to find the irumferene of irle. Mke r the sujet of this formul. 5. The eqution v u + t is used to find the veloities of ojets. () Mke t the sujet of this formul. () Mke the sujet of this formul. 6. The men of three numers, nd z n e found using the formul m + + z 3 Mke z the sujet of this formul. 7. Mke the sujet of the following formule. () v u + s () s t + 1 t CIMT, Plmouth Universit 18
F1.6 Mthemtis SKE, Strnd F UNIT F1 Formule: Tet 8. The volume of tin n is given V π r h where r is the rdius of the se nd h is the height of the n. () Mke r the sujet of the eqution. () Find r orret to deiml ples if V 50 m nd h 10 m. 3 9. A o with squre se hs its volume given nd its surfe re given V h A + 4h h () Mke h the sujet of oth formule. () Find h if A 4 m nd m. () Find h if V 50 m nd 10 m. 3 10. The re of trpezium is given () 1 A ( + ) h Write the formul with s its sujet. () In prtiulr trpezium. Use this to write formul tht does not involve, nd mke the sujet. h 11. () Averge speed, v m s 1, time, t seonds nd distne, d metres, re relted the formul d vt. Mke v the sujet of the formul. () Use the formul to find the verge speed for eh of the following performnes t the Beijing Olmpis in 008. Give our nswers orret to deiml ples. (i) Usin Bolt when he won the men's 100 m re in time of 9.69 seonds (ii) Usin Bolt when he won the men's 00 m re in time of 19.30 seonds. (iii) The Jmin rel tem for the men's 4 100 re in time of 37.10 s. Eplin the differenes in the verge speeds. CIMT, Plmouth Universit 19
Mthemtis SKE, Strnd F UNIT F1 Formule: Tet F1.7 More Chnge of Sujet This setion uses some further pprohes to rerrnging formule. Worked Emple 1 The period, T, of pendulum of length, l, is given the formul Mke l the sujet of the formul. T π l g First divide oth sides π to give T π l g Now the squre root n e esil removed squring oth sides of the eqution, to give T 4π Finll, oth sides n e multiplied g to give l g so the rerrnged formul is Worked Emple Mke the sujet of the formul T g l 4π T g l 4π 6 5 To void leving 5 on the right hnd side of the formul, first dd 5 to oth sides to give + 5 6 Then sutrt from oth sides to give 5 6 Finll, divide 5 to give 6 5 CIMT, Plmouth Universit 0
F1.7 Mthemtis SKE, Strnd F UNIT F1 Formule: Tet Worked Emple 3 Mke the sujet of the formul 1 1 q + First sutrt 1 from oth sides so tht the right hnd side ontins onl terms involving. 1 1 q Now omine the two terms on the left hnd side of the formul into single frtion, first mking the ommon denomintor. q 1 1 q 1 1 Now oth frtions n e turned upside-down to give Eerises or 1 q 1 q 1 1. Rerrnge eh of the following formule so tht is the sujet. () 5 3 () 8 6 () (d) 6 5 (e) 8 7 (f) 7 5 3 (g) p (h) q 8 + (i) r q 5. For eh formul elow mke the sujet. () q 4 () z () z (d) 3 (e) v 1 4 (f) r 5 π (g) p + 4 (h) r 1 3 (i) 3 + CIMT, Plmouth Universit 1
F1.7 Mthemtis SKE, Strnd F UNIT F1 Formule: Tet 3. Mke u the sujet of eh of the following formule. () (d) 1 1 + u () 1 1 1 u + 3 (e) 1 () u 1 1 1 p u 5 (f) 1 u 1 1 + u 3 (g) 1 4 + (h) r u v 1 1 1 (i) q 7 u 1 1 1 p u l 4. The formul T π gives the time for pendulum to omplete one full g swing. () Mke g the sujet of the formul. () Find g if l 05. nd T 14.. 5. The formul 1 1 1 + is used to find the fol length of lens. f u v () Mke v the sujet of the formul. () Find v if f 1 nd u 8. 6. If ll is dropped from height, h, it hits the ground with speed, v, given () v gh Mke h the sujet of this formul. () Find h if g 10 nd v 6. () Mke g the sujet of the formul. (d) Find the vlue of g on plnet when h 10, v 4. 7. A ll is thrown so tht it initill trvels t 45 to the horizontl. If it trvels distne R, then its initil speed, u, is given u gr () Mke R the sujet of the formul. () Find R if u 1 nd g 10. 8. When three resistors with resistnes X, Y nd Z re onneted s shown in the digrm, the totl resistne is R, nd () 1 1 1 1 + + R X Y Z Mke X the sujet of this eqution. () Find X if R 10, Y 30 nd Z 40. X Y Z 9. The volume of sphere is given the formul V 4 π r. 3 () Rerrnge the formul to give r, in terms of V. () Find the vlue of r when V 75. 3 CIMT, Plmouth Universit