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Ottawa Printing Co., (Llfmted), 3 and ^ 1898. rove Street. ^immmmmmmmmm- f -Jc'l ^ TO ■> ?SiiiSi»;*^^;SS':4"S: < »m < ' t \ Mk V'f PREFACE. ■K-',. Tt is valuable variety of useful exercises, is istined to inspire students with an ardent desire \r more extended mathematical attainments than ^ose acquired from a limited study of abridged smentary schooKbooks. With a view of pro- )ting intellectual progress, I have given many ^orems of great utility, witlfi the greatest fssibte variety of useful problems and their flutions, in a limited space. No two are alike, id from each, ? rule or formula may be deduced »r the working of similar questions. Those [ho have acquired a knowledge of algebra and sometry, will find these exercises really attractive a source of profitable recreation. This little work, containing elaborate solutions all its exercises^ comprehends more proposi- pns than the first four books of Euclid. It must ridoubtedly secure a wide circulation and meritori- is success. The principal propositions have ien contributed by the distinguished mathe- iatical correspondents of the Canadian Almanac id Journal of Education ; selected and solved their Math^mntical Editor, the Author. I. 2. 5- « 6. I i y* ; -« > ;M . ;rq-;ji«* i'MjI^it'^J ' " * g"" ' . " *" IMPORTANT MATHEMATICAL TlfOBLEMS. 1. Find the area of a triangle whose sides are, J^ /J^T J^T Ans.'j^. . ^ 2. The square inscribed in a circle : square in a simicircle */. 5:2. 3. The square inscribed in a semicircle : square in quadrant !'. 8 : 5. 4. If an isosceles triangle inscribed in a circle * have each of its sides double of the base, the squares described upon the radius of the f circle and one of the sides of the triangle, shall be to each otheir as 4 : 1 5. * 5. If r denote the radius of a circle, the side of the inscribed square will be r rJ^t and the side of the circumscribed square will be 2 r. 6. If a denote the side of a given square, rad^ of inscribed circle shall be }a, and radius of the circumscribed circle will be j/^J'' The rectangle under two sides of any triangle is eqiial to the rectangle under the perpendicular mmem9i^:\ifimtyhf.^'ys;iim,,iR-.?-im ***' Area of triangle EFB=;42; 15 — 2=i3=BE;.-.42-r 6J=:1 F M— 6.46154 ; GB^6.6As 11.2: 6.6 : : 6.46154 : BM : 3.80769; 13—3.80769 =9.19231=? EM. Then ^^9.192312 -f 6.46T542 =EF= 1 1,23615 ,-f r ■ length of the bisector. \22. Given one side of a triangle and the lines drawn from the angles at it, bisecting the other two sides, to find the sides. Let C E = i6; A E = i2 and C F=i5. j^CF = CO=io ;. OF = 5; ?^ AE = EO = 8, '. A = 4. On C E describe the triangle CEO and produce E O to A=i.2, ' and CO to F ' -r^-Jf**;^ =15. Join A C, F E and A F ; ' then A F is II C E and =i of it, =^8. In triangle A OF, 8 . 9 : : I : i}i diff. ot segts. A M and M F ; Z'4^^^ F, and ^j-^e = MA. ^42 -3^Ty2 == MO = 2.04538. Again, 16 : 18 : : 2 : 2j^ === diff. of segts, CH and HE; then CH-= g}iy and HE = 6^- 1J102 —g}i2 = HO = 4.09076 + 2. 04538 = 6.13614 = HM = AN. CH — AM = 9i — 3A = CN = sH' ^JHM2 — CN2 js/6. 136142 +5.68752 = AC = 8.36653x2 : .mnM i v iftmmtmmfi lO = 16.73316= CD. MF :-. 4fV; HE MF 23- = 6^ — 4t^ = PE = 2tV JPE2 +HM2 = »j3A^'+67r36r4^= 6.55743 = EF;;. E Px2 ' 13.11486 = ED. A pole 16 feet high is broken by the wind in D; in falling DC touches the ground 2}4 feet from base of the pole ] or the base of right angled triangle and sum of the hypotenuse and perpendicular are given, to find the sides. Letx denote AD ; AC = 16 ; AB = 2}4 ' a; AC = b ; DB :: b — x. Then x2 +a^ = b^ — 2 b2 '-a2 ; A AD r . 2 dx + «2 = ^2 , and X r 162 +25^2 2 b = 7.8046875 ; DB or 2 X 16 DC = 8.1953125. 24. If a line bisecting the vertical angle of a iri- angie, be divided into part?? wh»ch are to each other as the base to the sum of the sides, the point of division is the center of the inscribed circle. ••T ">«««"« w^ ?*■ 4 .,%: 1: 3t II 25. In any triangle, it is required to inscribe a rectangle whose sides shall bear a given ratio to each other.— Let ABC be the giv^^ triangle ;| AB=b;lcD=A, and side of recti < ± to base=rjic, a,nCj adjacent side = nXf n denoting the given ratio. Then ^ ; ^ [] nx : k := G H, one Side of the - X X :-:-X.- h-^n h required rectangle^ When side of the inscribed square is required. As h \h . . . i • ^ ^ \ • X \ ft — X X,zzz ~ J— ■ ' rh + b. 26. A farmer has a triangular field, the distances from whose three angles to the middle of the opposite sides are : no, 140, and 160 yards^ respectively. Required the area of the field ? Put AD :=d(iio), E C=i6o (^ ; B C = x^ AB =:v ; AC = ^.Then^ ^ +j/^ = 2^- -f JMr X2 ;jy2 ^^2 =z2{i' +^ Z^ ; Z^ -{-X^ =22'^ -{-% y^ The sum of these gives 2x^ + 2y^ + 2z^ — 2 b^ •¥ 2c^ '\' 2 d^ + % m^^ mmm^tmmmm 12 x2 +^jy2 + J^ z^\ andic2 4. ^2 4.^,2 «-.| ^2 -f I ^2 +1 rf* ; ivQm this subt. the first equation ; then jt^ = ^ rf^— tH^-^-^ c^— yi x^ , or gx^ = 8^2 _|. 8^'2 __-. ^2 J ^ ^ 2 = 81J2 -f 8^2 --. 4^2 ; whence x = ji J 8c^ + 8d^ — 46^ =C B = 186.547 + ' ' y = H JSi^ + Sd' — 4^2 . 2: '= J^ fj 8^"+ 8^2 >_ 4^2 • 186.547, 15748, 129.615 = Sides. 27. A stone is weighed in a pair of scales which are known to be incorrect ; when placed in one scale, it weighs 714 lbs ; but being pu' in the other, it only weighs 37 J lbs ; required its true weight. tJ7^\^37i = fJ 2700 = 51.9615. (nean pro- portional. i S¥*waw« .TT^'^i'«'%^f*^m4»<>'' 13 28. The base of a triangle is 80, and sides includ- ing the vertical angle are 65 and 55 perches respectively ; required the length .of a line drawn from a point within the triangle, 8.53. perches from the side AB, so as to cut off 4 of the area. CB = 65 = f ; BF = x ; ratio 5:7; AB = 55 = d ; w + w = J = 1 2 ; AC == 80 = //; BE = 9I = g ; EP = 58.67 = / ; AP = 8.53 : CG = 59.71 ; BH = 8. 53 ; EH = 3.67. By similar triangle s, we px have : g + x : p : I X BF : BA X BC g + x m : m + n. = BI. Then B p X2 ^+x : ^/ : : m SpX2 * J. Then bmf = -, and S ^ x2 ^x X ' ^ = bmfx4-bmfg .'. x = 32.617:5 = BF .'. Fl is easily found. . ■Aiuw^itdmSai ^il^tiifi^»^4m& 59. In a given triangle, the base AC = iod; A B = 70 ; BC = 90 ; uhat is the length of a - line (i) drawn 11 to the base. (2), x to the base (3), inclined to the base at a given < 15*^, so as to cut off t\ of the area ?— ag. i). Let X denote the length of the line 1 1 base AC : then (xix.vi. cr) 1002 ; x2 : 1 3059* 41 : yV of 3059.41. From this we get x = 7977 = QR- (II.) Area ol triangle B D C = 61.1882 X V = 2019 2106. Let X c required L, H L then, as B D» : x« :: B D C : y\ x 3059.4- 3744 : x« : : 2019 2106 : 1946.891^. From this proportion we obtain the value ofx = X to base. A C = 60.08258 r H L. what must be the sides of a similar triangle Ur 20 contatning 3^ times its area? Area of giv^n triangle = 336 : 336 x 31^ or as i : 3^ : : 26*; X* . *. X =46.87216. In like manneri we find AB r 50.4777 and BC r 54.0832*9. 38. The perimeter of a r < <^ triangle, is 74.4* and ^rom the r < on hypotheiiuse is 14,88 ; jSnd 0t, '^ne sides. ^^ — Let P = Perimeter ; CD = a; AC = X ; BC = y. "* ThenAB = p.-(y + y), 'and AC2 4. BC2 ? AB ; whence x^- -f + x2 + 2 xy + y^ transpose and divide by 2 and P (x + y)~J^ P2 r xy (i). By ^similar triangle s, AB : BC : : AC : CD .-. ABxAC, or a P-^a (x + y) = xy. (11). By^ Substitution (a-hP)x (x + y) =^ a P + JP^ ; whence x-f y r ^ P (a + ^P) or P(fl + JP) . « + P "' T ' — X. Substitute these vatues for a + P (x + y) and y in Eq. (11.); the result when simplified and reduced, givesi {a 4- P) x^ — P (rt + iP) x= -i flP2 . From last Eq and value P(« + :,P) + of y abpve is found, x or AC 2 {» + P) 21 P. / rw^ iJ(a-hP)^ -2a^' and if the result of 2(a+P) ^ * ' the two sides be taken from P, the result t3 will given AB«P p-(x+y)«-72 — •. which a(V*iii 11 22 of the sides ; 4 y* + 8 y r area of ends ; 12 y' + 36 y + 24 = area of floor and ceiling. 12 y* + 36 y + 24 - 508 = 10 y* + 14 y ; and y* + II y = 24a. Solved gives y = 11, the height ; 3 x x 1 1 + 3 r 36, the length ; 2 X 1 1 + 4 = 26, the breadth. W. D. 41. The wheel of a carriagfe is 5 feet in diameter, required the length of the successive cycloidal arcs generated by a nail in the circun\f<*rence of the wheel, in a distance of 24 miles ? By means of the calculus, we find the height of arc of a cycloid as follows ; J= 1^2 r x-^ -f vers — 'x, the equation pf the curve ; -4^ = dx t-x ^2 r x-x ■2 s _'J 2 r z - X ^J2 r X -ui? 4- *. ds :?= d X X dx S dx X J~ .'. S= 2 7-^9 a;-* dx = 2 fJYrx^ C. When a;=0, * S = 0, /. C = O, and when a; = 2 r, then the semicycloidal arc = 2 ^2 ^ ="4 ^ and the whole length of the cydoid is 8 r-4 times the diameter of the gener-. lii ^l_ 23 ating circle. . ' . ^^^ ^ 30-55767 miles 3.1410 travelled by the nail. - 42. There are two circles inscribed in an is Obceles triangle touching each other and the sides of the tri. angle ; • their dia- ■ * meters I are 4 and 6 respec- tively. Find the ' sides. From the centers DartdF draw D G and F ' H p e r- pendicu- lars to BC, and FO n CB : draw CFDA. Put DG = r:,FH=5: DO = y- — 5 = ^: FD^r-^s^-d. Then FO = Jd»— c» ^d; CB = a. By similar triangles D ■MHIMIl MiiiMiMiaiiUHiiiii '^('^'■ii ' ''*.•,''> -■■••,". •i,;V-,.:>^i.^.v:;:; A -■'■ \ 7-. ■' ■- ' ""-..- ^ ■ . - ■ ■:■■-. ! , ..-■. - 5 ■ ».„— ■*■ *c F O and B C A, we have b\ d\\ a : 1 1 I -V = A C ; and c : b o - r: — = CD, then ¥ r a — + ^r c d d ' 43- ■^- ': Find the ladius of a circle whose centre being taken in the circumference of another circle containing two acres, shall cut off one half its area ? Radius of first circle:=20. 18501 18. Now, suppose A O = 11.7 (nearly) ; t'hen 20.1850118 X = (11.7)^ ~ 136.89 ;. X = 6.7817615 = height of the lower segment. Then 11 .7 — 6 . 7 8 I 7 6 I 5 = 4. 9 182385 = height of upper - segment. Then the area of upper segment A C E is found = 65.7318402, and area of lower segment A O E :: 94- 3903972 : sum of segments = 160.1222374, too much. Again, suppose 11.69, ^"^ proceed in like manner we have : * . I ^ ■■.■'• ^- y .' ' v'm'"*', ■ -■ ,-' - '^s-rWi;' 5-^5.-1;. ':«^14'2^. - ;* . 1'' f. ■ ' - '' ' .''-»,-■'. l.v 65.7299442268 = area of upper segment. 94.1699210470 - area of lower seqfment. 159.8998652738, too little; error = . 1061347262 and .122237465. Using these errors by the rule of trta/ and error y the radius approx- . " imates'to 11.6944789, and simi of segments = i6o,ooo,ooo + a decimal. f 44. A line 16 inches long is a common tangent to ^ -;. 4 circles touching one another. It is divided in the points of contact A B C D, such that A B» = 3 B C« = 5 D . The difference be- tween the diameters of the ist and 4th circles is 7.311521963. Reqd. areas of the 4 circles. ' ■•ilfl'l "I "• I'\ Hi 26 A B^ =3 B O , and B O =5 D C^ .'; 3 B O = 15 D C^ , hence A B^ = 15 D O ; BO =5 C D^ , then A B = ^rsxDC: BC^^ixDC lAJTl^ J^^ I) X D C = 16 and D C = j-^n -f I = c. Then A B = ^^15 x z =» 8.71673739. B C = Jl X C D = 5.03261066992 .'. C D = 2.2506. Let ^jc. y, z, and u represent the diameters ^respectively : then x y^a^ =3^ =liii^^j8r =5 f^z /. sy-=5 ^» and j=*f ; x^3z .*. xi^=! c^ or u^ -f du = 3 c*' This Eq. gives w = >i fJd2 4. 1 2C2 = — - 1 .6^8478- 2 037;'S^=-~r''3-oa»i36j;:K=— =8.44»S5- 1418: x = -~ =0 then y 3. 0000 1 36 1 'x. 7854 = . . 7.q686 ~ areaof 3«'dclrcIe- I.688478*x.7854 r 2.2'i9i = area of 4th circle. 9*x .7854 63.6174 = area of i**^ circle. 8.442351 'x. 7854 = . . . .55.978 = area of 2nd circle. ■fai 45- 27 An aged man two daughters had, And they were very fair ; To each he gave a i^iece of land, A (^ircie and a square, At twenty shillings an acre just The land its value had ; The money that enclosed the whole Just for the land was paid. If every shilling be an inch. As it is very near, -^ Required the acres in each piece The circle and the square ? For the circle. — Let x denote the circumference in inches ; then o^ x J^d by =0795775 = area in square inches, and 6272640 square inches in an acre .*. — ' — ;; = price oc all. 6272640 ' X t. 3941214.54 r circumference .*. 394214. W X .0795775-6272646 = 197060 • 7338 = area of the circle. Forthe square.— 3941214.54-7-20 = 197060.727 = area as before for circle. Let x denote the side of square, then x^ = area in inches. x*- 6272640 = 4^ X r 1254528 inches. mmttnk 28 12545^^^ ^ 20 4 X X r 1254528. 6272640-^ 1254528 > 4 -i- 20 r 250905.6 = area of the square. * 46. A tre6 standing* in the water is just 15 feet ; above the surface. When the wind is blowing the tree is bent over and touches the surface 20 : feet from where it stood. Find the length of the pole or tree. Let the line A B represent tlie water ; C P the pole or tree. When P touches the water at B, C B must be equal C P .*. triangle B C P is isosceles — AB r 20 feet : AP = 15 .-. ^202+152 = 25 =PB .-. PD or DB r 12^^ ; 1 AE = 12, and PE rQ; then 9 : 12 : : i2j^ : 16^ = CD. Then ^CD2~7 PD2 = ^jr^>^^M6^ = ^P = 20J, length CP, as required. 29 47- ■A ■ 1*',' 48. ..■;«■. 4Q. 50- AD r 24 ; BC = 18 ; DE = EC ; required ED, AB, AE, EB and DC. It is. easily proved that triangles ADE and BC E are equian- gular, and D E = E C .-. they are = in every respect* and AE = 18 : BRTiif; ,-. M^^f^^ 30; DC =42.4264. The 1 from M, middle pt. of DC. finds pt. E. ^ '^ The parallel sides of a trapezium are 20 and 26 ; the area is 996 ; find th« ± height. — Area ^ by Ji sum of s'des = x height.w^!^^^r To find the area a rhombus. — Multiply yi of one diagonal -by the ^other. -!^^|t' PW^r0f • To inscribe a circle in a rhombus.— Intersec- tion of diagonals is the' center. ■■ .'■■■■■»*f#*-'' '■'.''V'-gK. ..-"••tw: '^ .*t;f,ii -•.' ..->v.- .,'- -*^' -^ . ;- .-'1 •'.^•ft'- |i. To find the X Jp* ^*^J from the ?- < on the - hypoteituse, when the sides are represented r^by 3, 4, 5,- or n time^y^ese number, sum of 3 sides??- 5 r x ft: 'n^'^^r^f '5 V 30 52. Find the diagonal of a cube, the length of whose side being 18 inches. T^,ei|Sqi«araillK>t 0^3 fftlMthe square of side A='t'e vft«rnm diagdnaL ; . 'he length of a rootn iis'^S.feet ; breath i3>i, ind io}4 feet high ; recttiired the distance from any angle of foor )f the ceiling ? ^7 e farthest corner = 22. 5 ; the required diag- onal. 54. Find the size of the largest square stick that can be cut from a cylindrical piece of wood, 5 j4 teet in circumference and 1 2 yi feet high. Inscribe a square in circle representing the base, and Inultiply its area by the height. 55. The sides of a trapezium in- scribed in a circle are 40^ 60, 80, and 67, to find the angles. Tri- angles ADO and B CO are similar.*. AD : * — ?l BC : : OD : OC ; for a similar reason, AB : DC : : AO"i OD : OB : OC ; hence we have the proportional lengths of AO, OB, OC, and OD that is AO = i.79tV ; OB = .37^^ ; OC = I, and OD = 2. The relative lengths taken two by two, give the rates AC*:«BD, and AC >^' BD== AB>^ CD + AD " BC, calling ^c, bd, the hypothetical values of the diagonals, we have bd : ac, :: 7220 : AC^ , or 2.89f| : 2.79y\ :: 72120 : AC^ .-. AC = 83.42 + , and 7220 — 83.42 = 86.54. We have now the 3 sides of each triangle to find he i% and BCD^ios" 9' 43". j6. Given the diagonal of cube, 20.78461 feet ; find the length of the side or edge. The ' square of the diagonal of a cube is equal '3 times the square of the side. 57. A cone 28 inches high is bisected by a circle || base ; how far from the vertex is this circle ? Let *= height of upper cone, cut off. Then 28* • Jt* :: 2 : / .-. 2jic3 = 288 ; x^ ~ io^y5 ^.^ X = 1^^ fJT =2242 in. ^59. Z]^ slfipt'heigfht otil cone is 12, and its solid ity 521.1537408 r required the height? mni 3a 2 1563 - 46r2224 4612224 .-. x* = jj^^-^ Let y deflate the hilg>it, aod i^ the diameter of the base ; then fjiz^-x^'^y^ »"^ 4 ^ ^ ^ .7854X? ^521.1537408- 3i4»6^2y ^,563. •. 144 — ■ 3.14107 '■'''-:::■}/■ 1S63 4612224 ^ 2, and y3- i44y --497. . ,3.Hi6y 664, solved, y = 9 • 6. the height. 59. A^piece of square timber is 12 feet long ; each side of the greater base is 1 1 inches and that of the less, 5 inches. What length must be cut off from the less end, so as to contain a solid foot ?. . . . AD = II .-. AE ' S'M BC is 5 ; BO = 2i. ThYoughBdrawBF ii HE • then AF r 3, and FE = 2^ ; AF : FB :: AE t EH. or, 3 : 144 :: si - 264 inches = EH ; but • EG = 144.*. OH = 120 inches. 5* »< 40 = lopo cubic inches ; iocKy+ 1728= 2728^ = soli- dity of pyramid KHL; u2 X H^ = 10648^ solidity ADH. LetKLt X ; then 11' ix^ '\r. r.i): ' ;*.'!;. v^'^;.\>'V-. *... -^ :.»-i. ■•■ ■■ ,:jt.i7'.-.-. -^■■iiiiiJ- ,j , r.l.^i' ;,^i_ 33 :: 10648:2728; whence x3 = 341 .-. x = 6.9863, length of the dividing line As 2J : 120 :: 3.4932 : 167.6736 = Pn ; .-. 167.6736- 120 = 47,6736 = PO. ^ 60. A bubble of air having a diameter of 4 inches, passes from ♦^he bottom of a lake to ti.' (op 120 fathoms. Required the diameter of the bubble on reaching the surface. 120X6X f2 — g — x62i =3121^ = pressure per square inch on the bubble. 42 x 3. 1416 = 50.2656 r surface of the Jdubble. 312J X 50.2656 = 15708 = pressure on bubble at the bottom. >tV%^ 5^*2^5^ = 3*636 : pressure of = size at •the top. As 3.636 : 1570S : : i : 4320 ; 1^74320 — 3. 1416 - II* 13- 61. Two men purchase a circular race course, i mile in diameter, and divide it by a line // diameter' ; required the length of the dividing y fence, so that one may have fi of the area, and the other ys ? The area of a circle whose diameter is i mile, is .7854 mile, yi of which is .2618 of a mile. The versed line answering to this area is •36753395 which taken from the diameter i. l^mmtMM M. r II l.llllli MW «U m 34 . * /^- •' *;i>f/- 4;>v leaves .63246605, remainder of diam. Then i§^ semichord is a mean proportional bet\veen 4S^ ,;■•••. -^^/ci-,- >.. }«., .■.;.'.,',:•.■ ' the segments of the diarn. 5?v we have li^^J; 2 ^^[63246605 X .36753395 = .96426162 mile ^ -. #7697 yards. ,. ,- ;.^:,-^,::.n^-i,i.rr'\:yy^'"-->>- 62. A can mow a field in 15 days by getting 7 „n days helph from B ; and B can mow it in 24 days by getting 2^ days help from A. In what time could both working together mow .:^,::..,^'-,:,;..^*.,.>^^^^^ Asi ^4 : 7 :: I : 5 A + B B ;yj> V, .'.I ofA=i|4 of BAs I : 2Jj^ : i || : 4 • o^ ■, K I i^X rrtimereg-.rfl" ' a ■.■ .'Vrt O U -■ •.-r'.- ■ ■ '.,.1. - V" ■ ■ - -■ .. .- >■-(■'■;.«: .-. ■''■/■";■- v« ,. - - .•- ■ ' - ^ ■'■ • V • ■■■' > *■* J* ' ■-st"'' '^'i"'-.,' - -V *■■-" ■. 63, Find the length of a band to surround two wheels, the distance between, whose centers '^ being 14 feet, and diameters 10 and 6 feet respectively. C N = 14 ; C P = 5 ; Q N = 3. When tangents are on alternate sides of the circles, on the iine joining centers, discribe a semicircle, and make C L^ = sum of the radii, r 8. Then ^""(14^ -82 > = 1 1.489125. DN.The »*1 ^^^'h:] 35 ^i':^r%:<- !j"' ^ii'^'..''i''."^i' ;1|;| triangles C D N and O H N are similar ; . S*-^^*^ 8 : 1 1. 4891 25 ;: 3 14.308422 r HO, r B O ; *. II. 489125 -4.308422 = 7.180703. : A O, ^... = G O. ^~ (A O'^ + A C2 ) = C O r r " (7.18070324 52) = 8.75. Then, in triangle CAO, 8.75 : 7.180703 + 5 :: 7.180703-5 : 3'^3S7^3 - <^^ff- of segments = O S - C S ; hence C S = 2.857144 ; and S O ^ 5^892856. V~(5' --2.857144' ) -- SA . SG r 4. 10325.^8= ^j^^~( 2. 1428552 + 4. 103252 ) r 4.6291 = P A or P G r chord of i the arc A P G, and 2 A S r 8.206512 5 chord of the whole arc A P G ; hence the arc A P G = 9.608762. Circunife- -■'■/ , 36 .,- .^. -_ rence of the whole circle = 31.415926,-9.608762 = arc ATG, = 21.807164 + 2 X 7.180703 = 36. 16857, length of band O A T G O. Again, AH-GO = OB or HO = 4.30842^. NB or NH = 3 ; 14-8.75 = ON =5.25. Then, in triangle OBN ; the base : sum of sides &c &c .*. 5.25 : 7.308422 :: 1.308422 : 1.82 1428 = diff. of thesegts. .*. OV =3.535714 ; NV r 1.714286; QV = 1.285714. ^~(BN2-VN2)= BV = 2.461955. (B V2 + Q V2 ) i = B Q = 2.7774^ = chord of i arc B Q H. Then, chord of whole arc = 2X2.461955 =4.92391. Hence, th^ arc B Q H may be found to be = 5.765225 .6 X 3.1415926 = 18.8495556. -5.76525 = 13.0843 r BZH, +2X4.308422 =21.70114 =belt OBZH, +36.16857 = 57.86971, the entire length of the band. - > 64. When the tangents are not on alternate sides : CN =14; CP =5; NO =3; PO =6; CD =2. Then, ^"^{14^- 22).= 13.855406= DN = AB = EF.^(DN2+DA2 ) = AN = 14.177446. In triangle CAN, we have the base to sum bf sides &c. &c. .•.14 .'19.177446 :: 9. 177446 : 12.571427 = N S - C S, and the sum = 14, ;•• N S = 13.285713, and G S =.714287. Then r-^. g. "fli^", wi««!«nniP>^n>^pa(;ii;i«|HVi(PHP|>Maf^p|^^ PiPi"PI 39 X denote the rate; then 317.0465 X 100 • 31704.65 -r X P. 1 00+x : X :: 31 704.65 -rx : 297. From this proportion, x = 6.7496633 .*. 31704.65 -^ 6.7496633 = 4697, the principle required. . . 68. -Two. railroad trains 109 and iii feet longf respectively, are moving on parallel rails; when they move in opposite directions, they pass each other in 2j^ seconds; but when they move in the same direction, the faster train passes the other in 15 sec. Find the speed of the trains. Let x = ft. travelled by the faster per second ; a >d y =^ ft. per second travelled by the slower train. Then, — ~ 4- x + y III 220 or = 2}^ = combined speed of x+y x+y '^ both; and 2^ x-f 2^ y = 220 .*. 5 x-f 5 y = 440, aj\d X H- y = 88. Again, — -^ H or = x-y x-y , 220 , / — « = 15 .-. i5x-i5y = 220, and x-y= 14^, We have now the sum and diff. .'. x ~ 51 ^» and y = 36^. .*. speed of faster per hour = 35 m, speed of the slower 25 miles. mm MiaaiMiM 40 69- A person changed a fifty dollar gold piece for 31 pieces of foreign coin ; some of which were worth $2.26 each; others $3.01, and the rest 77 cents each ; how many did he get of each sort ? Let x+y-f z = 3i ; 226X-I-301 y+77z = 5cxx) 77 x+ 77 y + 77 2 = 2387 I49x-h224y * -2613 and 224 y = 2613 — 149 x = a whole num- ber; x-y 224 = a whole number = P. If P = 0, x=i ; y=ii and z = 19. /. the required numbers are i, 11, and 19. 70. A cubic foot of gold weighs 1 1 cwt ic4 lbs, and a grain can be beaten out so thin as to form a leaf of 60.25 square inches; how many of these leaves will be required to form an inch in thickness ? , j . 144 : 175 :: iiio^ : 1350 Troy = 7776000 grains -r 1728 =• 4500 grains in a cubic inch ; but one grain covers 60.25 •*• 60.25X4500 = 271 125 leaves form an inch in thickness, and the thickness of a leaf is i -r 271 125. j7i. a vessel [A] contains 20 gallons of wine; another [B] 8 gallons ol water. How many gallons must be taken from each and poured ■l^MliHy Uilti^^i^Mltlka iidMiyb ■m' 41 into the other, so that after repeating the process any number t)f times, the quantity of wine in each vessel may be the same as after the firtt operation ? WINB WATER ' 20, and 8 ; Let x denote the quantity WATER reqd. x + 20-v, and 8-x+x =first*remain- , _, 19 x , 380 -r iQ X , 54-7 X ders. Then -^— and ^— and ^^-^ 20, 20, o, 7X 8 - X , X X 20 - X 20-x 8 • 8-x ^ X and ;r- . . . . 20, 8 20 26 20 -180 - 19 X . X ^ ^ « . ^^ ^— + o ; or, 8oo~4 X = 760-38 X + 20-0 5 X . • . 7 X = 40, and X = 54. . Otherwise. — Let X gallons to be taken from [A] ; y = number to be taken from [B] ; then 2o~x : y : : 20 : 8 ; but x = y . • . ^ =54. 72. An Arab tent is composed of canvas k yard wide and ^ inch thick ; and when not pitched is wound on a pole 2 inches in diameter, forming 115 rounds of cloth ; how many yards in the tent ? 2.04x3.1416- 6.403864 - 4.56x3.1416=^23^^ 20.73456 XII5 = 2384.4744 in. = 66.2354 yards. MttcMiiiaiMlHUilHMIMMilliMM If^- 42 Otherwise. — ns^^z - 4.6 inches = thickness of cloth ; then 2 x 4.6 + 2 = 11.2 = diameter of pole and cloth ; .*. (11.2)2 ..22 = 121. 44X. 7854= 95.378976 inches, surface of end of cloth ; then 95.378976^36 = 2.6494i6-T-^*y=66.23i4» 73. A grain merchant having a quantity of barley^ sold I of it at a certain gain per cent ; fi at twice that gain, and the remainder at 3 times the gain on the first lot. He gained upon the whole 30%. What was the^gain on each lot ? Let P denote the required price ; r = rate per cent. Then | x f^^^ = f ^^ = profit on the first lot. I - X T?F ~ l?f == profit on the second. . ' TT X \h = l?f = profit on the third. Their sum is fy^f = profit on the whole ; 29Pr _ 30 p . r = 29 r = 450 .• 1 5VI = profit % on the first. 31 j\ = profit % on the second. 46VI = profit % on the third. 74. If A had travelled -^ mile an hour faster, he would have finished his journey in JJ of the time ; but if he had travelled ^ mile an hour slower he would have been iJJ hours longer 43 on the road. How many miles did he travel ? Let X = miles travelled ; y = miles per hour. Then — = time he couldfinish;y -h _2 II 1 1 X _ 37 X or, 420 X y = II y + 2 39 y, '^ ^ "^ 407 X y H- 74 X, or, 22 X y = 74 X .*. 1 1 y = 37, and y = 3^^ miles. Again, y - tt : x II X X , 31 . ^ ^ 2 r^ * :: I : = - -f — . . 35x = 363y2 -66y J I y-2 y 35 "^^ o oj J = 3885 .*. »= III miles. 75. A broker has two kinds of money. It takes m pieces of the first to make a dollar, and n pieces of the second, te make the same sum. Seme one offers him a dollar for r pieces ; how many of each sort shall he take ? z Let X = pieces of the first ; and y = piece^ of the second. Then niY. -ny— i.oo. If m=Jy x= 100 ; ;;^ = 2 ;x = 5o ; m = 4, x = 25;. ;;^ = 5»x = 2o; ;;^=io, x=io; m = 100, x= I. Again, if ;^= ico, y=i ; n = 20,y = 5; ;/=io, y=io; ^^ = 5, y = 20. U r=:Sf then we have^ ;;^ = 5, and ;^ = 20. .*. 4x20+4x5 = 100; .*. he takes 4 44 twenty cent pieces, and 4 five cent pieces. 76. A, B, C, are 3 equal vessels ; the first con- tains water ; the second, wine and the third contains wine and water. If the contents of B and C be put together, it is found that the mixture is 1 1 J^ times as strong as if the con- tents of A and C had been treated in like manner. Find the proportion of wine to watpr in C. y'■•'>■■'^■- Let X denote de wine in C ; then C - x = -water in C. Then 1st mixture. Bjfx C -X = strength of \\r-__ = strength of sec- ond mixture ; ^'^^ = "^ ; .*. a + x-x^ I - X 2 - X -ii}i x-iij^ x^, and x^ - x ?= - jy •'• x = .73425 = wine in C ; I - .73425 = .26575 • = water in C. 77. The discount on P dollars due x years hence : $ P :: m : n, find rate of int. Let /'ssrate, and x = time ; then loo-f rx P'>* X -——J = discount on $ P for ^^, % -■Mi ''if; '■i> : rx : : P: *^- 45 X years ; ; : r : : m : n, or^ : I :: m : n: r x : loo+r x : : r X loo+r X m : ft] this proportion gives x = TOO (£) or r= 100 / m \ X I n-m. I 78. A steamboat started from Hamilton, and sailed down Lake Ontario. Owing to rough- ness of the lake, during the first four hours, she only sailed 23 miles ; of this, the third hour's sailing was double the first ; the fourth was i ^ miles less than sum, and the second was yi of the distance =18 miles more than twice the third hour's sailing. Continuing this progression, how far was the boat from Hamilton, at the end of the tenth hour ? Let x^Ist hour's sailing; 2x = 3rd; ==4th;4^i±iK=2nd. ^ 3 22 X = 7i>i ; x = 3J<. Then 3J<+4?<+6>^ + 8>^ + 10^ + i3J<-h 16+19+ 22K + 25^ = 130 miles. i ■teiita IM 46 79. A local superintendent divides $360 between four schools in G. P., and each receives as many dollars as there were pupils attending it on an average. N". i receiving the smal- lest share. It was then found that the.amount received by N" 3 exceeded J^ of that received by N" I, by $90. Required the 4sum appor- tioned to each ? Let x = first term, and r the rate ; then x-f rxH-r^x-f ^^x = 36o, and r^x-| = 90 .'. 4 r^x-x = 36o ; and (4x^~i)x36o, ^ndx = -^Then 36o . 360 . 4/^—1 4 x 2 _ 4r^— I. = 1 .•. /^ +;^ -f-r-f 1=4 /^ -I, and r^ - 3 ^ -!_;.= — 2, solved, x = 24 /! r^2\ hence the respective values or shares, are : 24, 48, 96, and 192 = 360. W. D. So. A, B, and C commence bunsiness with an aggregate capital of $1200 ; C*s capital exceeds A's by $100. A sells out in 10 months, B in 7^2 months, and C in 54 months, and each receives in stock and profit $700. Find each per.son*s stock. iiL.u^ ,JmA^ ..vOj^^,--^,,:.^-. ■.■...■>^!..:...:^...\..*.i--..;.^ ..■>.■. ^H^^d^i^iMiiriitaHiMiMMIiillMliH ■T *** S? ' ^ *,"^ f ■« ■ fT'"*^ "."^ f^ 47 Let x = A's capital ; ioo4-x = C's ; then IIOO-2 X=B's. XX io + (iioo-2 x) 7^2 +(ioo+x)x5«=5J^'^^50. tly, 12^+79250. ; consequen- 16200 X 10 X 900 9 .. , ' * x+ 15850 = x .*. 162000 x = x''^ 4- 15850 X .*. x = 35o = A's profit. Then 700 — 350=350== A's stock. 1100—700 = 400= B's stock ; x4-ioo = 45o = G's stock. J.C. 81. A miller mixes flower which cost him $5 a , barrel, with some which cost him only $3 a ^ barrel, and sells the mixture at $5.40 per barrel, making 42.5 per cent. Required the proportions of the mixture ? Let X = first ; y = second ; then 5 x 4- 3 y = first cost ; 5I (x 4* y) = selling price ; or 5I x4-5l y = (5 >^ + 3 y/ = l X + 2I y = pro- fit. As 5 x4-3 y : | x4-|y : : 100 : 42J. Whence, x= 15, y = 23, and the ratio is 15 : 23 or 23 : 15. 82. When two metals are mixed in equal volu- mes, they form a compound of specific gravity 9. When mixed in equal weights. jk x.r jjj^tij^tMiijMim ■lii 'J.iS?'?. ■■;■ ^^^" they form a compound of specific gravity 8^. Find the specific gravities of the metals. Let W, w, and w' denote the weights of the compound ; w, w' being equal. S, s, and s' = specific gravities. (S — s')S: -r(s-s')SxW-w;and^4^^§^ xW (s-s ) X S = w. Then ^-^^P^-~— X 10 = 5. From (s-s ) , r , this 4o(s4-s')=9 ss'. Again ^ 4- 5000 S _ 50D3 S 4- 5000 S . g _j_ g'^jg" s' 9 By substraction, transposition &c., 8ic,J we obtain s = 10 and s' = 8, the specific gravities reqd.- 83. Two men, A and B, take shares in a Petrolia oil well to the aniount of $1850. They sell out at par ; A, at the end of 2^ years ; B at the end of 7^, and each receives in capital and profit $1400. How much did each em* bark ? ^ Letx = A's capital; 1850 — x~B*s; xx 2>^ 4- (1850 -x)x7>i = 13875 -5 x = sum of products. 13875-5X : 950:: 49 2jx : . 4?5x _ 2775 A's profit; 13^75-$^ '■ 95o::i3875-7ix: ^^36250-i425x ^ ^*^ O /J /» 2775-X B's profit. As 950 : 13875 - ^ x : : 1406 _ ^ , 3885000-4175 x+x^ ^ ^^ j^^j ^f 190 . '^ A^s capital and time .'. we have,' (3885000-4175 x-|-x2)-r475 = A's capi- tal =x. From this equation, we obtain x= 1091.863755 ; and 1850- 1091.863755 B's capital. 84. A renner had a tank full ot alcohol, contain- ing 177 147 gallons, from which he drew a certain vessel full, and filled up the tank with water. He repeated this process 1 1 times, when he found only 2048 gls of alcohol left in the tank. Find the capacity of the vessel ? - The quantity left after the nth draw is (17714.7 -~ xV' fouad to be ^ '' ^' ^— ' 2048 ; .'. 177 147 177147/° ^ /# T/ -X = 1771471^X2 : and 177147- x= 1 18098 . • . X = 59049 - Yz tank. 85. There are two quantities, m*a.nd n whose . arithmetial mean is x ; . the geometrical ■* *•> ■JJ.kjiik.>M I I I'l III"-' so .-1 mean is y ; and tjie harmonic mean is sf; if x^y=3l, and X — z=«4twi» find «» and ^"^^ ^. r — *. ^^^ n. z. Putx=*«^; y=;«. Then 3I; and m-^n 2 m» 2 — .1 76 --4T¥Y' fJmu Then X - y = 3I and x - z = 4t^|t. x + y * 2jxy •f 7V, and x4-y= ^^+9T?T.*.wehave 4xy 4xy • 2 fJxy H- H- ^^FyHhyr'^"^^^""^' ^^^^' ""^ fractions, ^ ^— ^ 1 + 252 -« 250 Wx y .V500 xy-h 504 iJxy + 18141 = 500 xy+ 1800 ^^i 296 ,J5^=*i8i4| .•.5iN/xy='7» and,J^««li,aad xy-y ; x+y«=2V+' Tiswio. Now, we have the sum, and product given, hence x«9t, m^g\; y«V •*• «*•• - 86. The dtffcretiir between the quotient add divisor is 81339, atid sum of the divisor and m. ■ 7t «•»' 51 dividekid is 3007249* It is required to Biid :e&cn« '9i^ii^m^ ' Let X « divisor ; then 81 239 4^ x =* quo- tient; but(8i2394-x)xx*=dividefid,and 3007249— x« dividend ; /. x^ +81239X = 3006249 - x; whence x^ + 81240 x = 3G07249. Then X =* 37. 87. The time/ rate» principal, and gain at com^ pound interest are all equal. Required the time? Let X denote each ; p R^ ^ s ; per ques- . tion,^=x; r^-^^ R=i+-^; t = x; / X \x 2 X .•. X X j i X — - — I = 2x; divide \ 100 J by x; X Vx 1 4- = 2 ; by the nature 100 Wgs, we have x x ( \ 1 + 100 .3pro300.xx \^r--^ ipo 20000 3000000 X M = &c 3010300 x^ 3^ M .X- &c., 100 aoooo 3000000 6g|i 7j by reversion,x ij« 8.49824, the answ. fW^-^^-" f'^^'''fyT'^y''<^yr 5* • #■ ' - 88. In what time could $j 5 amount to the same, if placed at 6 per cent simple, and ^ per cent comp0ond interest. By a few trials, the time is found to be bet- ween 43 and 44 years ; then by the general rule of **Trial and Error," the answer is 43)4. 89. A merchant bought a cask of spirits ^£4^^ and sold ^ quantity ^exceeding three fourths of the whole by 9 gallons, at a profit of t5%. He afterwards sold the remainder at siich a price as to clear 667;. ,on the whole transac- tion ; and had he sold the whole quantity at the latter price, he would have gained 175 per cent. Find contents of cask. ' Note.— Henri Mondeau. the Shepherd of Touraine, a wonderful French youth of extraordinary powers of mental calculation, having visited Jersey in order to csitibit these powers, and when asked the foregoing question, answered it almost instantaneously as fottlows : He sells the^ first portion at a profit of 35 per cent, and the last at i75 P«* ^®"*» ^^^ gahied 60 per cent on the whole. The first profit is hssBi Aan the meitti profit Jg?^ 3^ per I '»>,. S3 cet3t» and theisecond ts greater by ii^ per cent I he has .*. sold 115 parts ot the, first against 35 of the second, that is, the first portion sold w^s \H of what the whole cost ; and the last -^ i but the first portion was |^ of cask and 2 gls. more ; and the difference between {^^ or }J and ^, is ^V • *• 2 gls. ,s ^ of cask . *. lao gls = whole. 90, Which is the best interest, at 6 per cent com- pounded annually on ^1000 for 20 years ; 5% compounded every instant ? Let X = 00 all the instants in 20 years ; ^=1000; thenA = « ^j -I- ^^ x = ^ (I^^y^^ x(x-I) ^r^ ^ X(X->I)X(X^2) X 1*2 X2 I '2 '3 X ~ &c., hence, since x = infini ty, x (x — i) I, and the series is. (I-f■I^-J^- &c.>== 1000 X 2.7828= icoox, X 1.2 1.2.3.4 Naperion base of logs = 27i8.28, at 6% it is 3207.18 ; diff. = 448.90 in favor of 6%. J. Ireland. 91. A cylindrical piece of wood is 12 feet long, and ^j4 ft. in diameter.^ Find the solidity ijMet ■. 54 of the greatest paralleloptpedon that am be cut out of it. Find the area of a square i whose dtagonat is 3j^, and multiply this area by the given length. 92. A farmer uses a roller, 4 feet 8 inches wide, and 2 ft. 8 in. in diameter. How many revolutions does it make over 7A 3R 25P ? 8809.32, the answer. 93. A cow is tithered with a rope sp as to graze over I A 35 P of pasture ; but the grass being insufficient to teed her, what additional length of rope will allow her the use of another acfe? Answer 16 yds i fooi. 94. Required the dimensions of an upright cylin- drical vessel, capable of containing 16 gal- lons, when the depth is equal diameter of the base? .7854x2 X X = 277.264 X 16. This equation gives x = 17.809 = height, or diam. of the base. 95. If into a cylindrical vessel whose inner diam- eter is 3 inches, we put as many wires of ^ inch diameter, as possible ; how much water can be afterwards poured in, allowing the height of the vessel to be 12 feet ? 96. • 55 Put radius of the cylindriciir vessel suf* and small radius = 4 ; thep the number of small circles that can touch the original . ■■ip S6 . from which 3 lines are drawn to the angftes, measure 2 J, 2, and 1} respectively, construct the triangle and find length of side. ^he 3 given lines are 1^, 2, and 2^ tespec- tively equal half 3, 4, and 5 .*. they form a r < ^triangle. Now draw a line A B = i J ; from A erect j.* A C = 2 ; join B C = 2 .** Oil A C des- cribe an equilate - ral trian- gle ACD. From D let fall the X DEon BA pro- duced ; and from C, draw C F at / in F. Join E F ; join also D B, and on D B describe an equilateral triangle DBG. Then DBG is the triangle required. It is easily proved* by the principles of equilateral triangles, that triangle D C G and D A B are equiangular .'. C G = A B = ij ; C D = 2, and CB=2^; .'. C is the point ^^ f 57 within the triangle G D B, from which the , lines C B, C D, C G, drawn to the y similar tri- angles H G «= BC; but HG IS the COS. a (30*), or COS. a X coeff. of fric- tion « friction .'. I2I.243$X .05= 6.062175 units of work destroyed. Or, the Normal reaction multiplied by the coefficient ot fric tion is the friction. . • . .866025 x .05=043301 ^5 X 140=6.062175, units of work destroyed .'. 70 w- 6.062175 « 63.937825 = units of work accumulated in the sledge at C .-. 63.937825 -.05=1278.7565, th^ answer. Formula W (h - base x coef. frict ^^) (70-6.062175^11) === 1278.7565. The followincr is another solution : f-- 59 The pressures : « :: base : 140.'. 140^^= 131.2435565 121.2435565 w, and jf 140 pres- sure. work of friction. 121.2435565 ^ I _ 121.2435565 i4<^^ 20 2800 70 w _ 121.2435565 140 280P work of gravity over friction. Then, V** r 64t >^ 1278.7564435 wx i4o -| yi ^ . ^^ units of work corresponding to this velocity w «63,937822i75w^ — =" 127S . 756435435* A. D - 9^. The frustum of a right cone is 24 inches in diameter at one end, and 16 inches at the other end, and 60 inches in height. At what ' height will half the solidity of the whole be found ? Answer : The segments of the line representing the height are : height of lower frustum = 24. 17 1 2847, and height of upper frustum = 35.8287 16. Limited space prevents an entire solution. 100. In a railroad excavation there is a rock to be cut, % mile long, 40 feet wide at the top, and 25 feet d6ep. A contractor agrees to 6o complete the work for $59888!, or $.175 per cubic yard. How wide must he leave the cut at the bottom'? Answer 16 feet. Let X denote breadth at the bottom. Then 40'f-jc 1000+22^ 2--— X25- iooo+25JKr X 1320 = 1320000 + 33000 Jf 2 X 27 cubic yardsofexcavation=» if x 660000 -^ 1 6500^ 27 000 ft 660000+ 1 6500 AT ,^ = 59888I . •. = 34222} ;from 27 this jf=i6. 101. The height of a cone is 41.088 ft., and diameter of the base 8.56 feet? Find the length of a straight line || to base, and diam- eter of a circle that will bisect the cone. Let 41.088 = A, and :r-the bisector. ; then 8.56«'x .7854x^— ^ = 788. 192274= solidity of the cone, and its J^ = 394.096137. Then 8.563 : jc* :: 2 : i ; hence Ji; = 6.78575' x 7854 = 36. 1 7550475 = bisecting circle . •. 394.096137 ■^3^' 1 7550475^ — ; whence ^'=32. 68. 3 ■^fl 6i 103. Any part may be cut off, if instead of the ratio 2:1, we use the ratio of the whole solidity : the part to b^ cut off. 103. Find the side of the largest cube that can be cut from a sphere 10 feet in diameter ? The solidity of the greatest cylinder that can be inscribed in a given sphere, is the revolution of the greatest inscribed cube, and the side • of the latter, is 1 10' /T 5-7735- 104. Three men stand on a plane, in the same straight line ; the first is 6 f t 2 ; second 6 ft ; and the third is 5 ft 9 inches high. The distance between i*' and 2"** is 10 ft. Find the distance between the 2^ and y^^ whei^ '■ / the tops of their heads are in the same straight line. A C=74 inches ; E N=72 ; and B 0=^69. 74 -: 69=5=-L C ; 72 - 6g=3=3==^S N or L P .*. 2 : 10 :: 3 : 15 = 8 D or E B. ■J ^ .i h- :■ '<&.': '•:-■. ■*■ HAXIMA AND fllNIMA. 105. Find the greatest rectangle that Can be inscribed in a given triangle. The greatest inscribed rectangle is when its height — j^ A of given triangle. 106. To find the shortest, line that can be drawn through a given point between two other ^ lines forming a right angle. Let TB and BV be the two indefinite straight lines, B the right angle, and P the given point. From P draw P M x hase T B, and draw PA || T B ; then P M i^ given = ^, . and P A==a=^M B. Denote T M by jf ; then X is found « ^fja ^ ; the twa points T and B being then giyen» we determine th^ direc- tidn and lenght of TV, ' 107. To find the greatesit trapezium that can be inscribed in a given semicircle. When the radius ==a, and height of trapezium =^ ; then x= X ~,. To find the greatest rectangle that can be inscribed in a given semicircle. The greiUest rectangle that can be inscribed in a semitclrcle is, when its height is equal the radius divided by the square root of a. r h\t^^ .. ! 1 i 64 I CI. The equal sides Qf an isosceles triangle are opened from the vertex ; what must be the length of a line joining their evtremities, so that the quadrilateral thus form d, may be the greatest possible ? Answer, 4 ^ = ^^a + 8 *» 7 ^' £'.■■• ^ . V- ■ - . ■ ( ii2. If the given triangle is equilateral, substitute ing a for b, the result is 4 at = ^^a ^. g «■ a • -a; or ^ a-a '2 a ,\ X - — and the maxi- mum trapezoid equals 3 times area of the ' original triangle. 113. Two cofds, one of 6, the other of 8 feet ia length are attached to a weight 100 lbs, and fastened at their extremities to hooks in the ceiling, 10 feet apart. Required the strian on each cord. The sum of the sides represents the whole strain on both sides ; and the strains on the sides are in the inverse ratio of the length of the sides. Then, as 10 : 14 :: xoo M40, the total strain ; .'. 14 : 140 :; 8 : 80 = strain on € ft cord. As 14 : 140 :: 6 : 60=^ strain on 8 ft cord. i \-..- -..j, 115. An iron wedge whose angle is 14", is driven into a rntlass of oak by a force of 125 R)s. What force is necessary to extract it? ' (3>'5o +7") W sin. 38^ 50' ^ 627057- ^' - -66976x135 :S4.72.Ibs, the force required. 116. A beam of oak i foot square, has its end firmly embedded in Masonry, from which it |^rojectS9feet ; to what height could a wall of brickwork, 2 feet thick, and resting on the 4)eam, be carried, without producing rupture ? A cubic foot of brickwork is equal to 1 12 lbs* n 66 ^.. Let a = natural length of the beam ; b its depth, and c its breath. W = — x ; s being the . 3 «' "Inoduius of elasticity. Then, w - — x 3 12x144 42 . 1. r ^-^ r ^4""«~ ~ pressure for every inch of 42 length of beam. Then 246 -^ x 108 = 26618 89 86 — — 5)s, sufficient to produce rupture. 9x2x i x 86 112 =20i6.*. 26618-^ -r 2016 = 13.203 feet high. 117. The rafters of a house are each 18 feet long and tied by a wrought iron rod 30 feet long, and section ^ square inch. What weight must be sus- pended from the vertical an- gle so as to break the rod ? Wx 30^-4 AD = horizontal pressure ; A D = 0,95 .'. WX30 67200 39-8 " 4 .*. Wx;Jt) > i; il 39.8x16800 .*. W = 39.8x^60 : 32288 K>S. 118. •i I 19. 120. IJ^vi ■jat^tiu^glitfiiliiiim^ttiitgbuuittit -WpUUfiV'^R^ 67 118. A bar of wrought iron suspended vertically breaks by its own weight ; what is its length ? The tenacity of wrought iron = 67200 lbs. Let x ■*' = length of ihe bar and n the area of its section; 67200 n = breaking weight of the bar. Specific gravity of wrought iron = 7.788 .*. 7788 r X 144x16 • u. i-u 7-788 nx = weight of bar ; ,\^—^ — x , « at = 672O0 n 2304 V ,% 7.788 X =67200 X 2304 ; X = 19880 feet, length of bar. 1 19. If the traction power of 97 lbs is required to draw the fore wheel of a carriage over an obistacle 6 inches high, what power will be required to draw the hind wheel over it ; the diameters of the wheels being 3)^ and ^% feet respectively ? As the required power varies ih tersely as the radii of the wheels, we have 4^4 : 3^ :: 97 lbs : 75.4 =the required power. 120. To what depth may an empty glass vessel, capdble of bearing a {> v^ssure of 216 lbs to the square inch, be sunk in watei' before it breaks? , .03616 lbs avoir du poids £ weight of one cubic inch of water, at a temperature of 60^ ; 68 then 2i6-f. 03616 =5973-45 inches.-. 5973.45 ■7-12 =497.7875 = the required depth. 121. If team of horses draw 3500 lbs, and the center pin is removed one inch from the center, how much will each horse draw ? If the center pin is moved one inch from the center to to the right or left, the horse draw* ing on the short end wiU pull about ^ more >. ' than the other. 3500 x-jV = ^75 lbs, differ- ence. 175 + 1750 = 1925 ; 1750 - 17s = 1575 ; hence the diff. of dratt = 350 lbs. ' -- 122. A owes B $455, payable in 14 years, viz, at the end of every two years $65. But he agrees ,, to pay him in 7 years by equal payments each ^ year, which B agrees to, and at the rate of 6% compound interest. What must be the annual payment? First, find the present - worth of the seven payments, which were at first to be made, which is found to be % $293.2583. Then find what annuity to con- tinue 7 yrs at the given rate $293.2583 v/ill purchase, which you will find to be $52.5^, the answer required. / 123; If a body weighs 60 lbs at; a distance of 3000 miles above the earth's surface, what ^m i"l"iH' I , r-. ,"v • 69 will it weigfh at 3000 miles below the surface ? Find the weight at the surface. Bodies weigh directly as the tnasis^, and inversely as the square of distance above the earth ; but they weigh directly as the mas^^^below the surface. Then the body is 7000 miles from the center, what will it weigh at the surface? less .•. 7000^ : 4000* ;: 60 lbs : 19ft. If a body weighs 19!^ lbs at 4000 miles from the cen- ter, what will it weigh at 1000 miles from the center ? less. ^ As 4000 : TOGO :: i9ff : 4f^ lbs, the answer. 124. A belt is 16 inches long, and drawn round a circle whose diameter is 4 feet, until it reaches a point (P) in the dia- meter produced. Find the distance FP, BP. Also, if the circumference is equal to the length of the belt, BPmust be equal to the arc BF, 4ndBP + B'P 70 = the arc B F B'. Let belt PBDCEB' = i6 ; 3.1416x4 r 12. 5664 -f 2 r arc DCE = 6.2832. Then 2^ + 6.2832 = BDCB';2^ + 2JC =9.7168, aLndv-\-x = 4.8584 ; which is an indeterminate equation. , ^ If xzi; zy z 3.8584 ; and^3.8584» +2« = 4.34590 = A P, .-. FP = 2.34594. FP + 4 = - C P = 6.34594 . •. ^6.34594 X 2.34594 = y = . 3.8584; 2j/ =7.7168; BDCEB' =2 + 6.2832 = 8.2832+7.7168 = 16. 125. Two men engage to excavate 100 square « yards, divided by its diagonal into equal parts of sand and rock. They work frorn opposite sides, each, at sand and rock, until they meet in a line parallel to the sides at which they com- menced.* Each must receive equal parts of the whole cost which is $100 ; the sand at 75c., and rock at $1.25 per yard. How much rock and sand must each escavate and ^^^ where will they ^^ 7J 2' fc^'-A '■^ ••■■'*■ meetr^LetBE =jerAE = io-:»;; then EB = EO, and AE=CF=FO. Side of Mpare = lo ; A B C is sand and B C D is rock. Then lO-f fy (io-a:) xH + lO 300 - 3JC' 500 - iooa: + SJC** ^ 300- 3^' 8 "^ 8 '\'> 8~' . 500- 100^+ icjc^ 2x' . 100 ^-5 a;*, _|-2 ^ 5 — =^-— H Q-^~^ ;hen- *•■•'■, ,.\r i'-Tf^/C'-Lin: ce we whave x^ - ^o x ^ - 200 : solved gives ^=4.38448; 10- a; r 5.61552. 126. If E and D be the points of trisection of the sides A B, AC of a triangle (nearer to A), and F the point ot intersection of C D and B E ; prove that the triangle B F C is half the triangle ABC, and the quadrilateral A D F E is equal to eithej ot the triangles C F E or B D F. On A B, side of triangle A B'C, make A D = ^^^^ A B, and make A E r ^ A C. Join BE, CD, E D and A F. The triangle B C F will be J^ of triangle ABC, and the triangles E F C, D F B, and the quadrilateral A D F E will be equal to i one another. For in similar triangles A D E and A B C, E D = J^ B C, and in similar tri- angles DEF and BCF, side EF = ^ FB ; but l|||fi|«fi^^l!i^»iP'w-i^i^'j *|i .u -r^^!^'-^^/ .'-r-'. '■ 7» triangle C E B, by construction, = ?^ triangle A B C or # A B C ; but th^ base E F - }{ BE.-, triangle E F C = J^ A B C and B CF r I or yi ABC. Triangles BCE and BCD on Same base are equal » take away common triangle B C F, and EFC and DFB will be left equal. It can be readily seen that the two triangles forming quadrilateral = E F C or D P B.—D. Scott, C.E. 127. The girth of a heifer is 6)4 feet, and length from the shoulder blade to the tail bone 5. 25 ; 6J^« =42.25, and 5x5.25 t 26.25'; multiply- ing this together, and dividing 1.5, gives 739.375 lbs the approximate weight of the heifer when dressed. A shorter method i%^ to multiply the square of the girth (baqk of the fore shoulder) by the length ; then multf- .,,..^ • ■ (, ,«.,'. t of barrels^ casks. . 73 ply that result by 7 and divide the product 'by 2^ ", '';f^ ^.rf^'. vr laS. To find how many bricks in a watl or buil- ding, multiply the length, height and thick- ness ^ in feet by 20. A brick 8 x 4 X 2 = 64 inches. Jo find the Ci __ juare one hJ& the sum of the bung and head diameters in inches, and multiply by the height In inches ; then_multiply by 8, and cut off the I ight hand figure ; this gives the cubic Inches which divided by 277 Ji( gives .the number of gallons, and divided by 2150.4 gives the number of bushels. 130! Required the contents of a barrel whose ^ iniddle or bung diameter is 22 inches, and diameter 18 inches, and 30 inches high? 22 4- J, j tS 4-2 =20, the average diam. 20 x 20 x 30 X fiB =9600 -T- 277 J D is common to the r r D and A F = is '*S ■MHIilMIIIHlii '34 75 • A D. The r < flf triangle ABE gives A E = A B cot. E ; hence d^ h cot. ^ a* Agait> r < d triangle CAD gives AC = A D cot C i hence X^-Vh - d cot. ^ a -h cot? y% a. Example. — From the top of a mountain 3 miles in height, the visible horizon apperired depressed 2^-/»i 3' 27" ;. Find the diameter of the earth and^ the distance of the boundary of. the visible horizon. Log. Cot. %. h = 0.4771 21 \ t. }i fi - 1 1. 71 1941 j sum = 2.189062 ; d = = i54.:^4; + ^=7961.3.901003. h = .> • .*, D = 7958 miles the diameter of the earth. 135. To construct a horizontal sundial. — In any straight line A B take a point O, and from O \, erect a x -0 T ; take O as center withe the chord of 60^ as radius and describe a circle F T S D. F S will represent the 6 o'clock line, and O T the meredian or 12 o'clock. From T on the quadrant arc, lay off T H = the latitude and join O H ; from T draw the X T P which will be the sine of the latitude. Make T G = T P, . and from G as center with iHilittilili y6 G T as radius describe the quadrant G T L and divide it into 6 = parts. Through each of these parts draw the lines G U, G V, &c., meeting- a tangent from T, indefinitely produ- ced in the points U, V, W, X, and Y. Through these points respectively, draw the lines O U, O V, O W, O X, and O Y, which will be the hour lines from 12 to 6, p.m. Make the arcs on the quadrant FT r respectively to {•/"'■'-..^i''-: :. ff- ■w fT^^ ^ *. '^i" :f 'Ui' -i'/ %^ . ■«' ' , fe. !^'- ;!,' ,\ .'?iv \, 'i-S rf •^'■.- "■■ii-.- -% ■■H-. these, and we have the hour Unes for the fore-, noon. 7 (a.m.) produced gives the hour liiie for 7^ (p.m.) and 5 (a.m.) gives 5 (p.m.) &c. To find half hour lines, divide the quadrant GTL into 12 r parts and proceed similarly. The quarter hours will be sufficiently appro- ximate by bisecting the half hours arcs. ^ The angle of the gnoTion or style, standing' j^" ly on the meridran Hn^. O T, must exactly touch FS at O, and the meridian line must ? ; coincide with the breath of the gnomon. The * < of gnomon must be equal to latitude pf the place for which the dial is constructed* To describe a south inclining vertical diaU we use the complement of the latitude ; then the construction is the same as the foregoing, lo describe a dial for the equator. Divide a circle into 24 equal parts and place a x style in the center. This placed on an inclined plane, having an angle equal the latitude of a given place, will show correct solar time. A dial engraved for a given lat., will show correct time by placing it on a wedge having an < = the diff. of the two latitudes. 136. The three sides of triangle are 18, 12, and 10. It is required to bisect it by the shortest 78 line possible. Describe an isosceles triangle A F G - }4 triangle A C B, having a common <^ C A B, by the VI. 15, and the base F G is the line re- quired. Bisect A B in E ; then A E r 10.3^2304 = A G or A F. Area of triangle A B C = 56.56854, hence A F G = 28.28427. Denote^ F G by x^ then we have the follo- . wing equation ; f^a' - x^ x ^ = s = 28.28427: this equation gives jc = 2.828427 x 2 = 5.656854 r F G, the required minimum line, 137. A, B, and C in partnership gain $1800. If we takeC'stimv^ from the sum of A*s and B's, 7 times the remainder will be equal to 1 1 times the sum of A*sand C'sdimished Hy B*s C's stock is to the sum of A'sand B*s stocks; as A*s time is to 6 times B*s time ; the sum 4Qi all their times divided by the sum of B's and C*s minus A's, equals 19 ; and 3 times the difference between the stocks of A and B, is equal to twice C's stock. Requlired each person *s gain, by simple proportion. 79 7 (A's + B's-Cs) time r n (A's + C's-B's): hence 7 :ii :: A*s + C's-B's ; A*s + B's-Cs; then f8 \ \ :: 2 A's : 2 B*s - 2 C's ; .*. 36 B's - 36 C's = 8 A's or 9 B's - 9 C's r 2 A's also,9 B's-1-9 C's : 10 A's, hence 18 C's =8 A .•• C's tkne = % A's time, and 18 B's = 12 A's .•. B's =9X I2-T-I8 =6 ; then A's =9, B's =6, and C's =4. Again, C's stock : A's + B's stocks ;; 9 : 36; from this proportion, we find C's = (A's + B's)-- 4 = (3 A's - 3 B's)-r 2 ; this gives B's stock = \ A's. Let A's stock rix9=9 \^5« ^^^o ;: 9 : 1080 = A's g:ain. ** B's •* =4x6 =44/ 15; 1800;; 44:5144 = B'9 gain. ** C's *' =fx4 = 14/ »5 J 1800;; 14:2054 = Csgr'n sum of products of S & T - 15 ■>. $1800 = sum of gains. 138. A and B are candidates at an election when 680 persons vote, and A is defeated. The same electors vote the following, year, when A and B are again candidates, and A is successful, having carried his election by i| times as many votes as he h-jfore lost by, and his majority ; B's the year before ;: 9 : 5 ; IBMdHi gUMaM mm m -,, So how many electors changed their minds during the year ? Let I lo denote B*s gain and A^s loss the first year ; then i lox i| r 198 = A*s gain the second year. Then 198 : no :: 9 : 5, and 198 - no =8l5 who have changed there minds at the second election. This question admits 36 solutions clear of fractions, giving as many sets of answers ; and the majorities change 36 times from 10 to 360 included. 139. A owes B $1000, and agrees to pay him in ten equal annual instalments, at a rate per cent, simple interest, equal to the TRUE equated time for all the payments : how much must B receive annually ? Let ^ =th# rate = time ; then ^' = int. on each payment, and 100 + ^' = each annual payment. The most correct method of find- ing the equated time is, when the interest of the i*ums payable before the equated time, from the times when they are due till that time, should be equal to the discount of the| sums payable nfter the equated time for the' intervals between that time and the times at which they are due. Then, when x is the xs. :"«& 8 1 equated time, the times for intefesntre - •^ -- 1, .f -- 2, X - 3, .1? - 4, and ^ -5 years. The times for discount are : 6 - *, 7 -- ^, 8 ~ •^% 9- *,andio~^ years. ^^ (100+ ^* ) X (^ - i) = AT* - A^* + 100 x"" - 100 Xi X X (100 + ,ir* ) X (at -- 2) r ;,>> - 2 .t3 + 100 AT* - 200 /r ; .tr X (100+ a:' ) X (a: -3) = A^ -3 x^ + IGO at* -300.^; :r X (lOO + Jic:* ) X (^ - 4) = AT* - 4 Ar3 + 100 JC^ -- 400 ^ ; :v X (1^0 + -^* ) X (^ - 5) ~ ^* -- 5^^ + 100 X' - 500 X ; The sum of these products = (5^ - 15 ^ + 5°o^' -: -Soo^} ^ Interest. As tco-\r6x-x^ : 6;itr-rjic2 ;: 100 -♦•a:^ : \ 6x3 u. 6oojir - a4 - - I ooji;^ As 100 4- 7a; -:v^ : yx-x' :; ioo + a:'^ : 7x^ 4- 700X - x4 - lOOX' IOv?4-7X--X» As ioo + 8x--x* : 8x--x* :: ioo-{-x^ ; 8x3 + 800X -X4~ lOOX^ ioo + 8x--x* As 100 + 9X -X* ; 9x-x» :: 1004.x* : 9x3 4 900X - x4 - ioox2 ioo + 9X"x2 As loo+iox-x' : iox--x2 ;: 100+ x^ ; iox3 + looox-x*-- ioox2 100+ I ox -X" CA •^^'T*** ' ,'%.!"■ ' '^- fiv^^ :■<} 'i,'^ ,'"( 's'"'...,- ' """.■:*? :/!'wi,^i;ii!!.jM iiv:;',i",i! 8a We now have — 6x^ ^ 600 - x3 -'^loox 7x « ♦ 700 - x^ - loox ioo+7x-x2 ioo-f»6x - x^ 8x2 . 800 100 + 8x -x^ x3 - loox .gx^ 900 - X3- 100 X IOO-1-9X -x* - i5x*2 + 500X,- i|oo ' ox^ " 1000 "X>^ "jgg^^^y^ 100 r lOX - X* 100 This equation solved, gives x - 5.29484 - rate .r time, and x« = 28.03533062565, the annual payment therefore, = 1 28.0353 i40.The sum of the squares of two numbers minus their sum is 14 ; and their product added to their sum r 14 ; find the numbers. xa +y2^(x4-y) r 14, and XY+ (x + y) ► 14. Denote x + y by u ; then x« 4-y2 = u + 14 ; and XY ri4 -^ u. Then 2 XY = 28 -- 2u ; X2-f2XY + y« =u-f I4H-28-2U = 42 - u, .-. x + y = J^T^, oru = ^42 - u • "* = 42 -u, .-. u2 -f u =42 : this equation gives u =6 =xH-y. .'. x2 -fy* z 20, and a xy = 28 -12 =16; .-. X2 - X Y + y* = 20-l6:£ 4 .-. x-y=2; andx4-y=6, hence xr4 and y = 2. ■ _^ . 143, Solve x^ ^^Ji ' i8,by a simple equation. 142. Find the compound interest of $80 for one month, at 6 per cent per annum ; for 2 months, 3 months, etc. 3i,- J^^S^W^S •^^>/