(9) If 240-625 c. ft. are dug out to make a well 2 ft. 11 in. in diameter, find its depth. (10) A cylinder is 21 in. in diameter and contains 9 c. ft. 1080 c. in.; find the length. (11) A cubic inch of brass is to be drawn out into a wire in. in diameter; find the length of the wire. [Solid content of wire 1 c. in.] (12) What length of wire 07 of an inch thick can be formed out of 77 c. in. of metal? (13) What must be the depth of a cylindrical tank to hold 3850 gallons of water, its diameter being 5 ft. 10 in.? (14) The height of a cylinder is 40 and its diameter 28; required the height of another cylinder whose diameter is 14 and solidity double the former. 157. The convex surface of a cylinder is its outer curved surface. 158. The whole surface consists of the areas of the two ends and the convex surface. Suppose A B C D to be a hollow cylinder of paper or cardboard. Let it be cut down at any one side, perpendicularly to the circumference of the base, as in E F, and opened out flat. It will be seen to form the rectangle E F F' E', whose length FF' is equal to the circumference of the base of the cylinder, and breadth E F equal to the height of the cylinder. Hence the area of the convex surface is found by multiplying the circumference of the base by the height or length. To find the whole surface, the areas of the circular ends must be added. 159. To find the area of the whole surface of a right cylinder. RULE. Multiply the circumference of the base by the height of the cylinder, and add to this the areas of the two ends. Ex. The diameter of the base of a right cylinder is 14 in., and the height is 10 in.; find the area of the convex surface, and of the whole surface. Circumference of base =22×14= 44 in. (Art. 99.) Area of convex surface = 44 × 10-440 sq. in. Area of two ends =2×14×14×4 = 308 sq. in. (Art. 102.) Area of whole surface = 440+308 748 sq. in. Ex. LXVIII. Find the area of the convex surface of right cylinders having the following dimensions : (1) Circumference of base, 4 ft.; height, 2 ft. 6 in. (2) Diameter of base, 18.2 ft.; height, 10 ft. (3) Radius of base, 8 in.; height, 12 in. Find the area of the whole surface of right cylinders having the following dimensions: (4) Circumference of base, 44 ft.; height, 5 ft. (5) Diameter of base, 16.8 ft.; height, 10 ft. (6) Radius of base, 2·52 ft.; height, 1 ft. (7) A cylindrical boiler 2 ft. 4 in. in diameter and 6 ft. long, is made of metal worth 6s. 9d. per sq. yd.; find its cost. (8) What surface of tin will be required to make a cylindrical vessel open at the top, whose length is 4 ft. 3 in., and the circumference of the top 3 ft. 8 in.? (9) The length of a round pillar of granite is 7 ft. 6 in., and its perimeter 9 ft. 2 in; required the cost of polishing the curved surface, at 3s. 6d. per sq. ft. (10) A garden-roller is 3 ft. long and 1 ft. 9 in. in diameter; how many revolutions would it take to go over a grass-plot half an acre in extent? THE PIPE OR HOLLOW CYLINDER. A D 160. Let A B C D represent a pipe, hollow cylinder, or cylindrical shell, such as a garden roller. Then it is evident that its solid content will be equal to the difference between the volume of a cylinder having its diameter equal to the outer diameter of the shell, and the volume of a cylinder having its diameter equal to the inner diameter of B the shell. The solidity may be also determined by multiplying the area of the ring formed by the outer and inner diameters of the shell by the length of the cylinder. This follows from the reasoning contained in Art. 149. 161. To find the volume of a hollow cylinder. RULE. Multiply the area of the ring by the length of the cylinder. Ex. The length of a hollow iron roller is 3 ft. 6 in., the exterior diameter 2 ft., and the thickness of the metal 1 in.; what is its solidity? External diameter =24 in.; internal diameter = 22 in. =14×24 × 24=4524 sq. in. smaller circle 11 × 22×22=380 sq. in. ring Solidity of roller =4524-380% = 72 sq. in. =1 c. ft. 1308 c. in. Ex. LXIX. (1) The exterior diameter of a metal pipe is 6 in., the thickness of the metal in., and the length of the pipe is 21 ft.; required the number of cubic feet and inches of metal in it. [To obtain the interior diameter, twice the thickness of the metal must be subtracted from the exterior diameter. Compare Arts. 18, 123.] (2) Find the volume of a cylindrical shell, the radius of the outer surface being 11 in., the thickness 2 in., and the height 14 ft. (3) Required the volume of a hollow cylinder, the height being 10 ft. 6 in., the radius of the inner surface 3 in., and the thickness of the material 1 in. [To obtain the diameter of the outer surface, twice the thickness of the material must be added to the diameter of the inner surface. Com. pare Arts. 18, 123.] (4) Find the weight of a cylindrical iron shell 14 ft. long whose outer circumference is 3 ft. 8 in., the metal being half an inch thick. [1 c. ft. of water weighs 1000 ozs., and iron is 7'6 times heavier than water.] (5) The greater diameter of a hollow iron roller is 1 ft. 9 in., the thickness of the metal 1 in., and the length of the roller 5 ft.; supposing a cubic foot of cast-iron to weigh 448 lbs., what did the roller cost, at 18s. 8d. per cwt.? THE RIGHT PYRAMID. 162. A Pyramid is a solid of which the base is any plane rectilineal figure, and the sides or faces triangles, of which the vertices meet in a common point, called the vertex of the pyramid. Like the prism, it is called a square pyramid, a triangular pyramid, etc., according to the form of its base. a Thus ABDCE represents square pyramid, of which ABDO is the base and E the vertex. The straight line EF drawn from the vertex E to the centre F of the base is called the axis of the mid. pyra A When the axis is perpendicular to the base of a pyramid, it is said to C be a right pyramid. The pyramids E F mentioned in this book are all right pyramids. B EF is the perpendicular height of the pyramid, and E G the slant height. EA, EB, E C, and ED are called edges of the pyramid, and are all equal. [When the height is mentioned, it refers to the perpendicular height, unless otherwise stated.] Suppose ABCDFGKH, a vessel in the form of a rectangular prism, to be filled K with water and to contain 3 pints. If now a pyramid of the same base and height be carefully dropped into it, one pint, i.e. one-third of the contents of the prism, will be found to overflow; thus show F A D H G ing that the volume of a pyramid is equal to onethird of the volume of a prism whose base and height are the same. Hence we obtain the following rule. 163. To find the volume of a right pyramid. RULE. Multiply the area of the base by the height, and one-third of the product will be the volume. Ex. 1. The base of a pyramid is a rectangle which is 18 ft. long and 12 ft. broad; the perpendicular height is 8 ft.; find the volume. Area of base-18×12=216 sq. ft. (Art. 50.) Volume of pyramid=×216×8=576 c. ft. Ex. 2. The base of a pyramid ABD CE is a square, each side of which is 12 ft.; the slant height E G is 10 ft.; find the volume. Since CD=12 ft., G F=6 ft. Also E G = 10 ft. Now the perpendicular height EF, the line GF, and the slant height EG, form respectively the perpendicular, the base, and the hypotenuse of a right-angled triangle. (Art. 70.) Hence, the square of E F the square of E G-the square of G F =100-36-64. and, E F=the square root of 64, or 8 ft. Area of base = 12 × 12=144 sq. ft. ... Volume of pyramid=×144×8-384 c. ft. Ex. LXX. (1) Required the solid content of a square pyramid whose height is 25 ft., and each side of the base 9 ft. (2) In a square pyramid the side of the base is 10 ft., and the perpendicular height 24 ft.; calculate the solidity. (3) Required the solidity of a pyramid with a square base of which each side is 8 ft., and the perpendicular height of the pyramid 7·125 ft. (4) The Great Pyramid of Egypt was 480 ft. in height, and its base was a square 765 ft. in length; find the number of c. yds. in its volume. (5) A pyramid of lead is 18 in. high, and stands on a square base 5 in. long on each side; find its weight, a c. in. of lead weighing 6.6 ozs. (6) The base of a pyramid is a rectangle 27 ft. long and 20 ft. broad; its height is 15 ft.; find the volume in c. yds. (7) What is the solidity of a triangular pyramid whose perpendicular height is 24 ft., and the sides of the base 1.3 ft., 8.4 ft., and 8.5 ft. respectively? [Find the area of the base by Art. 69.] (8) The height of a triangular pyramid is 16·5 ft., and the |