A note on Simple but Interesting Mathematical results as Inculsive Technology for Sustainable Solutions

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1 Ittiol Joul of Egiig Ts Tchology (IJETT) Volum 7 Numb - Sptmb 05 A ot o Simpl but Itstig Mthmticl sults s Iculsiv Tchology fo Sustibl Solutios. N.Shivkum Pofsso & H ptmt of Mthmtics R.V. Collg of Egiig, Myso o, Bglo-5009, Ii.S.Sih Pofsso & Cogitiv & Ctl Computig R.V.Collg of Egiig, Myso o, Bglo-5009, Ii Abstct I this pp w pst som simpl but itstig Mthmtics of impotc. Th sults :- A) th cicumfc of y ight cicul cyli is obti by multiplyig th squ of its imt by 0 th tkig th squ oot of th pouct. Wh th cicumfc of such cyli is multipli by / th of its imt w gt its. This will b vy usful s iclusiv tchology i cuttig sig of mtl sht without wstg i optimum m i fbictio iustis whv thy l with costly mtl sht lik bss o lumium fo mufctuig co o cyli typ of cotis fo multipl puposs B)Fomul fo fiig th sum to tms of sis.. C) Fom th squ of th sum of th ow th imt is subtct th squ of th imt. Th mi is oubl, givig th squ of th c its squ oot givs th msu of th c. This typ of sult will b usful i th fil of civil giig sigs. Thus, ths sults vy usful fo y to y poblms s Iculsiv tchology fo sustibl solutios. Ky wos: cicul cyli, imt, cicumfc, sis, c,. Rsult. Th cicumfc of y ight cicul cyli is obti by multiplyig th squ of its imt by 0 th tkig th squ oot of th pouct. Wh th cicumfc of such cyli is multipli by th of its imt w gt its. Lt us look t th followig figu mthmticlly. If is th imt of th bs/top, th th cicumfc of th top of th ight cicul cyli will b 0 0. If π 0 th this fomul gts uc to cicumfc of th cicl =, which is pstly us. A C B ISSN: Pg 7

2 Ittiol Joul of Egiig Ts Tchology (IJETT) Volum 7 Numb - Sptmb 05 Agi of th top (cicl) = 0 0 If & 0 th this fomul is Sufc of th cyli imt 0 imt imt cicumfc imt 0 imt Squ yojs Volum of th cyli of th bs pth of pit imt pth cub yojs 9 cubic yojs Not: Expltio of 9 0 N = N ISSN: Pg 8

3 Ittiol Joul of Egiig Ts Tchology (IJETT) Volum 7 Numb - Sptmb 05 Rsult. Fomul fo fiig th sum to tms of sis. H th us of icchå is tivil, lthough o c plc it i th fomul to ot pticul th, sy ( ),... Th fomul is giv by S = [{(umb of tms - icchå) (commo iffc)} + {(icchå ) commo iffc} + (fist tm )] o. of tms wh =,,3,,5,. i.. S = [{ ( ) ( ) }] wh = No. of tms, = commo iffc, = fist tm, = to = icchå, This is th sm s th usul fomul, s [ ( ) ] I o to fi out th msu of th ctl th squc-o hols of ll th th s, 5 is th fist tm, 8 is th commo iffc, 9 is th umb of tms. This is to b ustoo wll. Th quisitio i xcss of th umb of tms is multipli by th commo iffc; out of th pouct, th quisitio i xcss of uity s multipli by commo iffc is subtct. Whtv is th mi, to it is twic th fist tm, th it is multipli by hlf of th umb of tms. This givs th sum of th sis. Lt th umb of tms = Commo iffc = Fist tm = Icchå = =,...7 Accoig to th bov vs th fomul is giv by ISSN: Pg 9

4 Ittiol Joul of Egiig Ts Tchology (IJETT) Volum 7 Numb - Sptmb 05 ISSN: Pg 30 O B A Q h - h P Cho hight of sgmt S totl 7 7 Now it is giv i th vs tht =9, = 8, = 5 Thfo totl S fo ll th sv ths. Rsult3: Msu of th c Fom th squ of th sum of th ow th imt is subtct th squ of th imt. Th mi is oubl, givig th squ of th c (hu¼) its squ oot givs th msu of th c. Th bov vss hv th followig mthmticl ifomtio. Th fomul fo fiig out th cho of cicl wh th hight of th sgmt imt of th cicl giv. Lt b th imt, h b th hight of th sgmt, '' b th ius; i.., Accoig to th vs, th Cho. h This my lso b witt s Cho = h

5 Ittiol Joul of Egiig Ts Tchology (IJETT) Volum 7 Numb - Sptmb 05 Hc, by Pythgos thom OB OP PB OB PB OB OP o PB OB OP AB h Th sco vs givs th fomul fo fiig out th c itcpt by th cho of cicl, wh gi th hight of th sgmt th imt of th cicl giv. Th fomul fo fiig out th itcpt c is c h Th squ of cho of th bow (c) btw th (two coscutiv) cil poits (i th fom of ofouth of th cicumfc) of th cicul isl lso is twic th squ of hlf of th imt. Wh this squ is multipli by fiv ivi by fou th squ of th bow (c) is obti its squ oot givs th msu of th c.th mthmtics is s follows : If th imt is tk s th cicumfc s c, th ius s '', th (cho of qut c) = Alttivly, yoj C A Cho AB AC CB c B Cho qut ISSN: Pg 3

6 Ittiol Joul of Egiig Ts Tchology (IJETT) Volum 7 Numb - Sptmb 05 Th ltio btw th fig. qut c th cho hs b show by th utho s follows - (qut cicumfc) = Cho of th qut c 5 = by tkig squ oot qut cicumfc s 0 Coclusios Thus w hv s som simpl but itstig Mthmtics of impotc. Th sults :-A) Th cicumfc of y ight cicul cyli is obti by multiplyig th squ of its imt by 0 th tkig th squ oot of th pouct. Wh th cicumfc of such cyli is multipli by / th of its imt w gt its. This will b vy usful s iclusiv tchology i cuttig sig of mtl sht without wstg i optimum m i fbictio iustis whv thy l with costly mtl sht lik bss o lumium fo mufctuig co o cyli typ of cotis fo multipl puposs B)Fomul fo fiig th sum to tms of sis.. C) Fom th squ of th sum of th ow th imt is subtct th squ of th imt. Th mi is oubl, givig th squ of th c its squ oot givs th msu of th c. This typ of sult will b usful i th fil of civil giig sigs. Thus, ths sults vy usful fo y to y poblms s Iculsiv tchology fo sustibl solutios. Rfcs ) Jyhvlå with commty Vol I, Pt I, Smput 7, Pu¹pt Åcåy Bhýtbli Åcåy, publish by Ntiol Istitut of Pkit Stuis Rsch, Shvblgol, Ktk, Ii, 005 ) Loyvibhåg (Lokvibhåg), Åcåy Simh Sý¼i, Eit with Hii tsltio by H.L. Ji & A.N. Uphy, Sholpu, 9. 3) Aupm Ji: Cotibutio of Ji Åcåyås to th vlopmt of Mthmtics, M. Phil. Pojct Rpot, Mut Uivsity, Mut, P 5, 980. ) Philosoph Mthmtici Ytivâbh, Aht Vc, (Io) (), P 7-, 988., 5) Suvy of th wok o i th fil of Ji Mthmtics, Tulsipjñ, Lu, Rjstå, Ii (-3), P (3-), 983. ) "Philosoph Mthmtici Åcåy Ytivâbh som of his Mthmticl postults Pt. Jgmoh Ll flicittio Volum, Rw, P 30-33, ) "Cotibutios of Ji Åcåys to th vlopmt of Mthmtics" Ph.. Thsis, Mut Uivsity, Mut ) Aupm Ji N. Shivkum Mthmticl Cotibutio of Ytivâbh " Aht Vc (Io), 7( -3) Ap-Sp. 005 P (3- ), ) A.K. Bg, Mthmtics i Acit Mivl Ii, Chukhmb Oitli, Vsi, ) Bij Moh "Histoy of Mthmtics"Utt Psh Hii Gth Acmy (Luckow), 95. ) pk Jv N. Shivkum"O th pculi fomul fo fiig th sum of Aithmtic pogssio i Ii" ) ipk Jhv Pmvthmm "Th Msutio of Coch i Acit Ii"Aht Vc (Io), () P (3-5), 00. 3) B.B. utt A.N. Sigh "Histoy of Hiu Mthmtics; vols. Fist Eitio, Motill Bsis, Lho Rpit Combi Eitio, Asi Publishig Hous, lhi, 9. ) N.K. Ji Chuhi, Vic Ji Mthmtics Puim Sgh, Tlkhi, Ngpu,00. 5) M.B. Ll Agwl,. Sis i Ji Mthmtics, Åcåy hmsg Flicittio volum, Clcutt, P 9-, 98.b. Cotibutio of Jiåcåyås to th vlopmt of Mthmtics Astoomy, thsis, Ag Uiv, Ag, 97, P-7. ) R.S. Ll, vlpomt of Thoy of Sis i Ii, Allhb Uiv, Allhb. 7) S.S.Lishk S..Shm, Itouctio to thsis o Ji Astoomy. Ji Atiquy; 30 () P-9 8) T.A. Sswti,. "Th mthmtics i fist fou ", Joul of Ggth Rsch Istitut (Rchi), 8, P 7-5, 9., b. "Gomty i Acit Mivl Ii" Motill Bsis, lhi, ) A.N. Sigh, Histoy of Mthmtics i Ii, 5(I), P -53, (II) 55-9, 99. 0).E. Smith, Histoy of Mthmtics, ow Publictios, Nw Yok, 95., Histoy of Mthmtics Vol-, P-57. ) C..N. Siivsig, Histoy of Acit Ii Mthmtics, wol pss, Clcutt, 97. ) N.Shivkum, 009, Ph thsis, ptmt of Stuis i Mthmtics, Uivsity of Myso, Ii. ISSN: Pg 3