. An elementary course of infinitesimal calculus . or the same value of the altitude. 9. If a right circular cone be circumscribed to a givensphere, its volume will be a minimum when the altitude is doublethe diameter of the sphere. Shew also that the semi-verticalangle will be 19° 28 [= sin |]. 10. The right circular cone of greatest surface for a givenvolume has an altitude equal to ^2 times the diameter of thebase. 11. From a given circular sheet of metal it is required tocut out a sector so that the remainder can be formed into aconical vessel of maximum capacity; prove that the angle of t
. An elementary course of infinitesimal calculus . or the same value of the altitude. 9. If a right circular cone be circumscribed to a givensphere, its volume will be a minimum when the altitude is doublethe diameter of the sphere. Shew also that the semi-verticalangle will be 19° 28 [= sin |]. 10. The right circular cone of greatest surface for a givenvolume has an altitude equal to ^2 times the diameter of thebase. 11. From a given circular sheet of metal it is required tocut out a sector so that the remainder can be formed into aconical vessel of maximum capacity; prove that the angle of thesector removed must be about 66°. 53. Geometrical Applications of the DerivedFunction. Cartesian Coordinates. We have seen (Art. 32) that if i|r denotes the anglewhich the tangent, drawn to the right, at any point of thecurve y = H«=) ? (1). makes with the positive direction of the axis of x, thendy _ dx = tan -^Jr. .(2). With the help of this formula, several magnitudes connectedwith a curve may be expressed in terms of x, y, and dyjdx. Y. Fig. 36. 122 INFINITESIMAL CALCULUS. [CH. Ill If the tangent and the normal at the point P meet theaxis of a; in r and G, respectively, and if M be the foot ofthe ordinate, then TM is called the subtangent and MGthe subnormal. Hence we find subtangent = TM = MP cot ^Jr = y dx (3), subnormal = MG = MP tan ^ = y ^ (4), tangent = TP = MP cosec •^ normal = PG = MP sec ?>jr = y <m .(6). Again, the intercepts of the tangent on the coordinateaxes are OT = OM-TM= as-4-dy dxOU= = y-x^ .(7). (8), Ex. 1. In the parabola y = 4aaj we have, differentiating both sides with respect to x, and omittingthe factor 2, di/ ^di^^ •(9), which shews that the subnormal is constant and equal to 2aAgain, the subtangent is .(10), y^^=|l = 2x dx la and is therefore double the abscissa; in other words, the origin0 bisects TM. Ex. 2. In the hyperbola we have dx (11).(12). 53] APPLICATIONS OF THE DERIVED FUNCTION. 123 Hence the formulae (7) for the i
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