204 



KNOWLEDGE • 



[Aug. 18, 1882. 



centre of prav 



lb. r.'« , r. Let A M = 2 r. f o that the 

 uvcr i;. where G A = J. A M - 1. 



Then the man's weijiht is proportional to the cube of x, and may 

 be reprcsenteJ by I'ur*. (The m will not trouble ns.) And tlu- 

 moment y cf the man's weight around the centre line of the boat's 

 Icnirth is 



or y-mjr'(l-r). Since AB = lft. 



Hence, following first rule given in the last lesson but one. 



^-3m«(l-r)-fiix'. 

 dx 

 This represents the rate at which y increases as wo increase x. 

 When V is as prcat as possible, there is no longer any increase, 

 ilicreforc we mast equate the above expression to zero, giving 

 3mi(l-i) = mx', 

 or 3(l-T)-l 



3x = 2 



hi^. 



lUst be tho amoant of thwart occupied by the oarsftnan 



effective. 



• a man occupjang 



ft. of thwart would weigh 6 stone, our oarsman should weigh 



-ay 



-lbs. 



Wo see that whether for a maximum or miuimnm wo must equate 

 ^ to zero. Tho question itself will sl.ow whether u mailinum or 

 dx 

 minimum exists for tho deduced value of .r. 



One more example will close tho present lca.son. 



PaoBLKM III.— .4 sphere has a radius r. What is the greatest ri,jht 

 cone which can be inscribed in the sphere? 



128,, 



.3^- lb. 



Problem 11.— A person is in a boat B (Fij. 4) three miles from the 

 e.:re>t point A, nf a straight »h«reline, C D. He uishrs to reach E, 



I oinl 5 miles from A, as quickly as possible. lie can walk 5 miles 

 n hour, but onlj row 4 miles an hour. Where must he land ? 



B 



Suppose F is the point where he should land, and call A F, 

 Then B A is perpendicular t o C D. 



And BF-v/TiA' + »'=v /9 + »'. 



Uencc time taken in traversing B F is .i — -— ' 

 Again FE — 5— a:, 



Lot B D (Fig. 5), tho height of iho cone A B C, bo «•. Then D C, 

 tho radius of the base, is a mean proportional between B D and 

 1) E. That is I) C = ^x (2r-»). 



II once tho area of the base 



= ,rx (2r-..). 

 and the content of tho cone 



= ..(2r-,)x? = Z:|.'(2r-.). 



(I assume a knowledge on tho reader's part of the relations between 

 tho content and surface of cones, cylinders, spheres, and so on.) 

 Now put V for content of cone ; that is, put 



and lind the differential coellicient of ly according to Uio lirst rule. 

 We can write it down at once, thus, 



^ = i5* (2r-x)- — 

 dx 3 ^ ^3 



(the reader will at once see that tho two portions of this value aro 

 obtained by Rule I., in last lesson but one.) 



Now so long as by increasing x, y increases, wo havo not a 

 cone. Ilence, since tho differential coefficient expresses 

 dy 



and time taken in traversing F E is — — 



Thus if »- total time occupie d in r eaching E, we have 



6 = -l- + — • 

 Now following the two rules, we get readily, 

 dy„ X _1 



This cxprcrfiscs tho rate at which the time increases as F is moved 

 away from A. One can see that when x is very small the value of 

 — is negative, since a negative increase is decrease. This shows 

 that oar man will get to F more quickly by landing to the right of 

 A (cloac by) than actually at A. And as long as -^ continues nega- 

 at as z increases, we reach at last 



■0. 



the rate of increase, we must havo - 



= 0, 



X 



irx' 



(■2r-x)^x\ 

 4r.<.-3x'. 



(i) 

 vhero it is bIiow 



a value for which -— ceases to bo negative and becomes poaitivc, 



<U 

 passing through tho value nought. When r has that valne, the 

 rh'trfrnmi! hnn refi/'h«l its utmost ; so that to obtain tho maximum 



j^,. ..„..;,.. w.. \ -,- .. r.nly to solvc thc cquatlon, 



X 1. 



4 v'9 + a»~6 



„r, 51-4^9+7', 

 i.<-., 23x'-]C rO + jJ) 



This gives a I* -144 

 3z-12 

 *-4. 

 So that our traveller must land four miles from A or one mile from F. 



This gives .i-^> and, therefore, D must not 1 

 in Fig. 5, but D E must be equal to twice D. Tho reader will 

 notice that (i.) is also satisfied if x = 0. We see tliat when ic-=0, tho 

 volume of tho cone is also nought. This is a minimum not a 

 maximum value. It is clearly quite as necessary that the diftor- 

 ential coefficient should vanish to give a minimum as to give a 

 maximum. In all cases like the present, and indeed in nearly all 

 the most useful simple applications of tho calculus, tho conditions 

 of the problem itself show us wlien wo havo a maximum or a 

 minimum. There are rules for analytically determining this ; but 1 

 shall not trouble tho reader with them. lie sees in this casc^ 

 that tho content of the cone starts from when tho height is 0, to 

 again when tho height is 2 r ; hence at some part of tho passugo 

 from to the cone must havo a maximum value ; and tho abovo 

 process show.s him (what ho could not readily find by any other) 

 that tho cone has its maximum value when D E-2 O D. 



Wo regret that a very easy problem wits given at p. 172 instoad 

 of this : — 



To integrate ''""(^'iTl)''^" 



Wo regret that tho omission escaped our notice in proof (or the 

 index may have broken off in i.riiiting). Tho problem as actually 

 given was correctly solved by J. K. C, An Engineer's Son, Kit, 

 Menelans, Arbuthnot, and others. X. Y. Z. solves correctly, 

 another problem yet, having written 4v' in numerator. Iho 

 problem which should have been given (as above) is by no moans 



OuK WiiiwCoLUMK.— " Five of Clabs" considers that duing tho 

 holiday season, Whist on alternate weeks, wUl satisfy our readers. 



