be derived from the Prismatoid Formula? Which could be proved by the use of Cavalieri's Theorem, and how? Upon what power of the lengths of sects do area formulas depend? volume formulas? If the solids are similar, what rule holds for the area and the volume? If the line ratio is 3, what are the area and volume ratios? If the volume ratio is 3, what are the line and area ratios? If the area ratio is 3, what are the line and volume ratios? State all the ways in which spherical angles can be measured. What relation is there between polyhedral angles and spherical polygons? Explain why. How is this relation used? What is meant by "distance on a spherical surface, and why? Explain how a spherical triangle can have three right angles, and tell what its polar triangle is. How are symmetric triangles proved equivalent? GENERAL EXERCISES 303. If the strength of material is in proportion to its cross section, what is the effect of doubling the diameter of a wire? If a wire of in. radius will support an iron ball of radius 2 in., how large a wire will support a ball of radius 4 in.? What effect has doubling the diameter of a tree on the amount of wood in it? 304. How many tangents can be drawn from an external point to a sphere? How could one be constructed? How could a plane be drawn from an external point tangent to a sphere ? 305. Find the locus of the centers of all spheres that are tangent to a given plane at a given point. 306. Find the locus of the centers of all spheres of given radius that are tangent to a given plane. 307. Find the locus of the centers of all spheres that are tangent to the faces of a trihedral angle. 308. A cylindrical hole of radius 6 in. was bored through a sphere of radius 10 in., the axis of the cylinder being a diameter of the sphere. What was the weight of the remaining part of the sphere, if its entire weight was 35 lb.? What ratio had the total area of the remaining figure to the area of the sphere? 309. To inscribe a cube in a given sphere. 310. To inscribe a regular tetrahedron in a given sphere. 311. To inscribe a regular octahedron in a given sphere. 312. How large an iron shell 1 in. thick would a sphere of radius 3 in. make if melted and recast? 313. A cup is in the form of a segment of a sphere of internal radius 3 in., with a flat base of radius 2 in., and a cylindrical top of radius 2 in., and height 2 in. How much water will it hold? (231 cu. in. = 1 gal.) 314. Why is a plumb line, in general, perpendicular to the surface of the earth? 315. In mixing paint, it is important to know of what sizes to grind the spherical particles of coloring matter in order to cover the surface evenly. If spheres of radius r are tangent to each other and to a plane, find what size other spheres must be so that they can lie between each three of those spheres and be tangent to them and to the plane. 316. If spheres of radius r are piled up in pyramidal form so that each one rests on three others that are tangent to one another, show that the centers of the outside spheres lie in the faces of a regular tetrahedron. Find the height from the highest point to the ground if cannon balls of radius 6 in. are piled three deep. 317. If the edge of a cube is doubled, what is the effect on its area? on its volume? If the area is doubled, what is the effect on its edge? on its volume? If its volume is doubled, what is the effect on its edge? on its area? 318. Prove that the shortest sect from a given point to the surface of a given sphere is along a line to the center. (Two cases.) 319. Circles are inscribed in, and circumscribed about, an equilateral triangle. Find the ratios of the areas and the volumes of the solids generated by revolving the triangle and the circles about an altitude of the triangle as an axis. 320. If a right circular cylinder of altitude equal to its diameter and a right circular cone whose slant height equals its diameter are circumscribed about a sphere, show that the total area of the cylinder is the mean proportional between the areas of the cone and the sphere, and that the volume of the cylinder is the mean proportional between the volumes of the cone and the sphere. 321. How many square inches of gold leaf are required to gild a sphere 6 ft. in diameter, no allowance being made for waste? 322. A manufacturer of marbles uses a sheet of white glass 3 ft. square and 1 in. thick, a sheet of green glass in the form of a cylindrical surface of element 2 ft., thickness in., and length along a line perpendicular to the elements 28 in., and a piece of yellow glass which when placed in a tank 3 ft. deep that is a circular frustum of radii 2 ft. and 3 ft., raises the water from 1 ft. deep to 2 ft. 3 in. deep. How many marbles in. in diameter can be made by mixing these glasses, and what proportion of each kind of glass will be in each? 323. If all possible lines are drawn from a point to a sphere, the product of two sects from the vertex to the surface on any one secant equals the product of the sects from the vertex to the surface on any other secant, and any tangent is the mean proportional between the sects of any secant. 324. To construct a sphere of given radius tangent to a given plane at a given point. 325. Find the locus of a point such that the ratio of its distances from two given points is constant. 326. In filling a measure with fruit or vegetables of approximately spherical shape, would a larger quantity be obtained with those of lesser radius, or with those of greater radius? Why? 327. If cannon balls of 1 ft. diameter are piled four deep so that each one rests on four others that are tangent to each other, find the height of the pile. 328. A cylindrical fire extinguisher is 8 in. in diameter and 2 ft. long with a spherical segment 3 in. high on its end as a great circle. How much liquid is required to charge it? How long will it take to empty it by a hose in. in diameter if the liquid flows through the hose at the rate of 25 ft. per second? GENERAL SUMMARY OF THE FORMULAS OF SOLID GEOMETRY 317. MEANING OF THE LETTERS USED IN THE FORMULAS h = altitude, the perpendicular from the vertex to the base, or between the two bases. s = the slant height in a regular pyramid or its frustum, the element in a cylinder, and in a right circular cone or its frustum, the edge in a prism. b, p, r: when a figure has but one base, b is its area, p is its perimeter, and, if it is circular, r is its radius. b1 and b2, P1 and P2 1 and r2, m: when a figure has parallel bases b1 and b, are their areas, p1 and p2 are their perimeters, and, if they are circular, r1 and T2 are their radii; m is the area of the midsection. and r, are the perimeter and radius of a right section. Prt I. AREA. (1) Lateral Area. (a) Frustum of a pyramid = 3(P1+P2) (derived from trap ezoid). 2 Applies to the frustum of a regular pyramid, the entire pyramid (p2 = 0), the frustum of a right circular cone, the entire cone (P2 = 0), the right prism and the right circular cylinder (P1 = P2) For figures with circular bases, it becomes πs (r1 + r2) TS (b) Pyramid = P (derived from the triangle). 2 Applies to the regular pyramid and the right circular cone. For the cone it becomes πrs. (c) Prism = sprt (derived from the parallelogram). Applies to the prism (including parallelepiped), and cylinder. For the cylinder it becomes 2 πï‚¡s. (2) Area on a Sphere. (a) Sphere = 4 πr2 (rotating a semicircle about its diameter). (c) Lune = A 360° (42) where A is the angle of the lune (derived from its ratio to the surface of the sphere). E (d) Spherical polygon = (4 π2r) where E is the spher II. VOLUME. 720° ical excess of the polygon (derived from the fact that a spherical triangle is equivalent to half the lune of its spherical excess). Applies to any solid whose bases are in parallel planes, if it is bounded laterally by planes, or by curved surfaces generated by revolving a straight line sect or an arc of a circle about an axis. = Applies to the prismatoid, prism, cylinder, pyramid (b2 = 0), cone (b2 0), frustum of a pyramid, frustum of a cone, sphere (b1 = b2 = 0), spherical segment, spherical wedge (b1 = b1 = 0). Does not apply to spherical cone, spherical sector, spherical pyramid. h (2) Frustum of a Pyramid or a Cone = (b + b + √byb2) (derived from the difference of two pyramids). Applies to the frustum of a pyramid, the entire pyramid (b2 = 0), the frustum of a circular cone, the entire cone (b2 = 0), a prism and a circular cylinder (b1 = b1⁄2)⋅ For circular figures it becomes (r12 + r22 + rï31⁄2)· πλ (a triangular prism is composed of three equivalent triangular pyramids). For the cone it becomes πr2h (derived from the rectangular parallelepiped). For the cylinder it becomes πr2h. |