77. If the volume of a prism whose altitude is 9 ft. is 171 cu. ft., find the altitude of a similar prism whose volume is 50 cu. ft. (Represent the altitude by x.) 78. Two bins of similar form contain, respectively, 375 and 648 bushels of wheat. If the first bin is 3 ft. 9 in. long, what is the length of the second? 79. A pyramid whose altitude is 10 in., weighs 24 lb. At what distance from its vertex must it be cut by a plane parallel to its base so that the frustum cut off may weigh 12 lb. ? 80. An edge of a polyedron is 56, and the homologous edge of a similar polyedron is 21. The area of the entire surface of the second polyedron is 135, and its volume is 162. Find the area of the entire surface, and the volume, of the first polyedron. 81. The area of the entire surface of a tetraedron is 147, and its volume is 686. If the area of the entire surface of a similar tetraedron is 48, what is its volume ? (Let x and y denote the homologous edges of the tetraedrons.) 82. The area of the entire surface of a tetraedron is 75, and its volume is 500. If the volume of a similar tetraedron is 32, what is the area of its entire surface ? 83. The homologous edges of three similar tetraedrons are 3, 4, and 5, respectively. Find the homologous edge of a similar tetraedron equivalent to their sum. (Represent the edge by x.) 84. State and prove the converse of Prop. XXVII. 85. The volume of a regular tetraedron is equal to the cube of its edge multiplied by√2. 86. The volume of a regular tetraedron is 18 √2. Find the area of its entire surface. (Ex. 85.) (Represent the edge by x.) 87. The volume of a regular octaedron is equal to the cube of its edge multiplied by √2. BOOK VIII. THE CYLINDER, CONE, AND SPHERE. DEFINITIONS. 540. A cylindrical surface is a surface generated by a moving straight line, which constantly intersects a given plane curve, and in all its positions is parallel to a given straight line, not in the plane of the curve. Thus, if line AB moves so as to constantly intersect plane curve AD, and is constantly parallel to line MN, not in the plane of the curve, it generates a cylindrical surface. N F B D M E The moving line is called the generatrix, and the curve. the directrix. Any position of the generatrix, as EF, is called an element of the surface. A cylinder is a solid bounded by a cylin drical surface, and two parallel planes. The parallel planes are called the bases of the cylinder, and the cylindrical surface the lateral surface. The altitude of a cylinder is the perpendicular distance between the planes of its bases. A right cylinder is a cylinder the elements of whose lateral surface are perpendicular to its bases. A circular cylinder is a cylinder whose base is a circle. A plane is said to be tangent to a cylinder when it contains one, and only one, element of the lateral surface. 541. It follows from the definition of a cylinder (§ 540) that The elements of the lateral surface of a cylinder are equal and parallel. (§ 415) PROP. I. THEOREM. 542. A section of a cylinder made by a plane passing through an element of the lateral surface is a parallelogram. F Given ABCD a section of cylinder AF, made by a plane passing through AB, an element of the lateral surface. To Prove section ABCD a □. Note. It should be observed that, with the above hypothesis, CD simply represents the intersection of plane AC with the cylindrical surface, and may be a curved line; it must be proved that it is a str. line || AB. Proof. AD and BC are str. lines, and . Now draw str. line CE in plane AC AB; element of the cylindrical surface. (§§ 396, 414) then, CE is an ($$ 541, 53) Then since CE lies in plane AC, and also in the cylindrical surface, it must be the intersection of the plane with the cylindrical surface. Then, CD is a str. line || AB, and ABCD is a . 543. Cor. A section of a right cylinder made by a plane perpendicular to its base is a rectangle. Proof. Let E', F", and G' be any three points in the perimeter of base A'B', and draw EE', FF", and GG' elements of the lateral surface. Draw lines EF, FG, GE, E'F", F'G', and G'E'. Now, EE' and FF' are equal and . Then, EE'F'F is a □. (§ 541) (?) Then, base A'B' may be superposed upon base AB so that points E', F", and G' shall fall at E, F, and G, respectively. But E' is any point in the perimeter of A'B'. Then, every point in the perimeter of A'B' will fall somewhere in the perimeter of AB, and base A'B': = base AB. 545. Cor. I. The sections of a circular cylinder made by planes parallel to its bases are equal circles. For each may be regarded as the upper base of a cylinder whose lower base is a O. 546. Def. The axis of a circular cylinder is a straight line drawn between the centres of its bases. 547. Cor. II. The axis of a circular cylinder is parallel to the elements of its lateral surface. Given AA' the axis, and BB' an ele- B' ment of the lateral surface, of circular Proof. Let BB'C'C be a section made B by a plane passing through BB' and A; then BB'C'C is a . Then since BC is a diameter of BC, and (§ 542) (?) BC and B'C' are equal, B'C' is a diameter of OB'C', and passes through A'. Hence, AB and A'B' are equal and . . Then, ABB'A' is a .. AA' || BB'. 548. Cor. III. (?) (?) The axis of a circular cylinder passes through the centres of all sections parallel to the bases. PROP. III. THEOREM. 549. A right circular cylinder may be generated by the revolution of a rectangle about one of its sides as an axis. B Given rect. ABCD. To Prove the solid generated by the revolution of ABCD about AD as an axis a rt. circular cylinder. Proof. All positions of BC are || AD. (§ 402) (§§ 421, 419) Whence, ABCD generates a rt. circular cylinder. |