dicular to the height of 7 ft., the upper part forming a right cone whose slant height is 10 ft.? [Here we are to find the curved surface of a cylinder whose base is 14 ft. in diameter and height 7 ft. (Art. 159), and also the curved surface of a right cone whose base is 14 ft. in diameter, and slant height 10 ft. (Art. 169).] (24) A circular pillar of granite whose length is 30 ft. and perimeter 66 ft. is cut away at one end so as to form a cone, whose height is 14 ft. Find the cost of polishing the curved surfaces, at 3s. per sq. yd. [In this case also we are to find the curved surfaces of a cylinder and a cone. The height of the cylinder will be 30-14=16 ft.] THE SPHERE. 170. A Sphere or Globe is a solid bounded by a curved surface, every part of which is equally distant from a certain point within the figure, called the centre. The radius of a sphere is the straight line drawn from the centre to the surface. The diameter of a sphere is the straight line drawn through the centre and terminated at both ends by the surface. Suppose A B C D, a cylindrical vessel whose height E F is equal to the diameter C D of the base, to be filled with water, and to contain 3 pints. If now a sphere of the same diameter and height be carefully dropped into it, two pints, i.e. two-thirds of the contents of the cylinder, will be found to overflow, thus showing that the volume A B D of a sphere is equal to two-thirds of the volume of a cylinder whose height and diameter are the same. Now the volume of a cylinder is equal to 1x diam. × diam. x height (Art. 155), and since the height of the cylinder and the diameter of the base are equal, this may be written, 1 x diam. × diam. × diam., or K Ex. LXXIII. Find, in square feet and inches, the area of the curved surface of right cones having the following dimensions:Circumference of base, 14.5 ft.; slant height, 12.8 ft. Circumference of base, 5.25 ft.; slant height, 8 ft. Circumference of base, 3 ft.; slant height, 18 in. (4) Diameter of base, 5 ft. 3 in.; slant height, 2 ft. 8 in. (5) Radius of base, 2 yds. 1 ft.; slant height, 5 ft. (6) Radius of base, 2 ft. 4 in.; slant height, 6 ft. (7) Circumference of base, 7 ft. 4 in.; perpendicular height, 4 ft. (8) Diameter of base, 14 ft.; perpendicular height, 5 ft. 3 in. (9) Radius of base, 1 ft. 9 in.; perpendicular height, 2 ft. 4 in. Find the whole surface of right cones having the following dimensions: (10) Circumference of base, 8.8 ft.; slant height, 6.25 ft. (13) Circumference of base, 18 ft. 4 in.; perpendicular height, 7 ft. (14) Diameter of base, 4 ft. 8 in.; perpendicular height, 16 ft. 3 in. (15) Radius of base, 5 ft. 3 in.; perpendicular height, 23 ft. 4 in. (16) What will be the cost of painting a conical spire whose slant height is 120 ft., and whose circumference at the base is 63 ft., at 2d. per sq. yd.? (17) The slant height of a marble cone is 6 yds., and the diameter of the base 23 yds.; find the cost of polishing the curved surface, at 1s. 3d. per sq. ft. (18) The radius of the base of a cone is 24 ft., and the perpendicular height 84 ft.; find the area of the convex surface. (19) The conical cap of a tower is 10 yds. 1 ft. 6 in. diameter at base, and 7 yds. high; find the expense of covering it with lead, at 12s. 6d. per sq. yd. (20) A conical spire is 66 ft. in circumference at the base, and 14 ft. high; find the cost of covering it with lead, at 1s. 6d. per sq. ft. (21) How many yards of canvas, 1 yd. wide, will be required to make a conical tent 30 ft. high, and 17 ft. 6 in. in diameter ? (22) What length of canvas, 33 yds. wide, is required to make a conical tent whose slant height is 27 ft., and diameter 49 ft.? [See Art. 56.] (23) How many yards of canvas, yd. wide, will be required to make a round tent 14 ft. in diameter, whose side is perpen dicular to the height of 7 ft., the upper part forming a right cone whose slant height is 10 ft.? [Here we are to find the curved surface of a cylinder whose base is 14 ft. in diameter and height 7 ft. (Art. 159), and also the curved surface of a right cone whose base is 14 ft. in diameter, and slant height 10 ft. (Art. 169).] (24) A circular pillar of granite whose length is 30 ft. and perimeter 66 ft. is cut away at one end so as to form a cone, whose height is 14 ft. Find the cost of polishing the curved surfaces, at 3s. per sq. yd. [In this case also we are to find the curved surfaces of a cylinder and a cone. The height of the cylinder will be 30-14=16 ft.] THE SPHERE. 170. A Sphere or Globe is a solid bounded by a curved surface, every part of which is equally distant from a certain point within the figure, called the centre. The radius of a sphere is the straight line drawn from the centre to the surface. The diameter of a sphere is the straight line drawn through the centre and terminated at both ends by the surface. A Suppose A B C D, a cylindrical vessel whose height E F is equal to the diameter C D of the base, to be filled with water, and to contain 3 pints. If now a sphere of the same diameter and height be carefully dropped into it, two pints, i.e. two-thirds of the contents of the cylinder, will be found to overflow, thus showing that the volume C F B D of a sphere is equal to two-thirds of the volume of a cylinder whose height and diameter are the same. Now the volume of a cylinder is equal to 1x diam. × diam. x height (Art. 155), and since the height of the cylinder and the diameter of the base are equal, this may be written, 1 x diam. × diam. × diam., or K volume of a sphere having its diameter equal to the inner diameter of the shell from the volume of a sphere having its diameter equal to the outer diameter of the shell. We thus obtain the following rule. 175. To find the volume of a spherical shell. RULE. Subtract the cube of the inner diameter from the cube of the outer diameter, and multiply the remainder by [To find the outer diameter, twice the thickness of the shell must be added to the inner diameter. To find the inner diameter, twice the thickness of the shell must be subtracted from the outer diameter. (Compare Art. 18).] Ex. The outer diameter of a spherical shell is 10.5 in., and the thickness of the shell is 2.1 in. Find its solidity. Here, the inner diameter will be 10.5 in.-4.2 in. = 6.3 in. The cube of 10-5 is 1157-625; the cube of 6.3 is 250.047; 1157.625-250·047=907·578. Hence, volume of shell = 11×907·578=475·398 c. in. Ex. LXXVI. Find the solid contents of spherical shells whose external and internal diameters are respectively: : (4) 8.4 in. and 6.3 in. (6) 23.1 in. and 21 in. (7) How many c. in. of lead will be used in making a spherical shellin. thick, whose external diameter is 7 in.? (8) What is the weight of a spherical shell 10.5 in. in diameter and 1.05 in. thick, composed of a substance of which 1 c. ft. weighs 216 lbs. ? (9) Find the weight of a spherical shot of iron, 7 in. in diameter and 1 in. thick, supposing a cubic inch of iron to weigh 4 ozs. (10) The exterior diameter of a hollow sphere is 21 in., and the thickness of the shell 4.2 in.; how many c. in. are there in it? (1) Given that a cubic inch of iron weighs 44 ozs., find the of metal in 224 spherical shells 11 in. in diameter and 'ck. ind the volume of a spherical shell whose outer diam. 11 in. and thickness 1 in. Ex. LXXVII. Miscellaneous Examples on the Mensuration of Solids (1) How many 3-inch cubes can be cut out of a 12-inch cube? (2) A rectangular block of granite is 5 ft. 6 in. long, 4 ft. 6 in. broad, and 2 ft. 8 in. thick; what is its cost, at 4s. 6d. per c. ft.? (3) A piece of timber is 20 in. broad, and 16 in. thick; at what distance from the end must it be cut, so that the part sawn off may measure exactly 5 c. ft. ? (4) The diameter of a rolling-stone is 18.2 in., and its length 4 ft. 6 in.; required its solidity. (5) What is the solidity of a sphere whose diameter is 14.7 ? (6) The depth of a canal is 7 ft. 3 in., the width 20 ft. 4 in., and the length 10 miles; how many cubic feet of water will it contain? (7) The length of a cellar is 27 ft., its breadth 16 ft., and its depth 8 ft.; if the cost of digging be £2, what is the price per c. yd.? (8) Find the solid content of a square prism whose length is 48 ft., and its breadth and depth each 12 ft. (9) What will be the expense of dressing a conical spire, at 8d. per sq. ft., the circumference of the base being 44 ft., and the slant height 45 ft.? (10) Find the surface of a globe 5.6 ft. in diameter. (11) Find how many cubes whose edges are 4 in. may be cut out of a cube of which the edge is 8 in. ? (12) A room 18 ft. long and 12 ft. broad is flooded with water to a depth of 6 in.; find the weight of the water. (13) A pint contains 343 c. in.; how many gallons of water will fill a cistern 4 ft. 4 in. long, 2 ft. 8 in. broad, and 1 ft. 1 in. deep? (14) A reservoir is 26 ft. 8 in. long by 12 ft. 9 in. wide; how many cubic feet of water must be drawn off to make the surface sink 1 ft.? (15) What is the solidity of a spherical ball whose circumference measures 5 ft. 6 in. ? (16) A block of stone 4 ft. long, 21 ft. broad, and 14 ft. thick weighs 27 cwt.; find the weight of 100 c. in. of the stone. (17) Find the cost of re-gilding the surface of a globe whose radius is 1.75 ft., at d. per sq. in. (18) A cube contains 5832 c. yds.; find the length of a side. (19) A triangular prism has the three sides of its base 2 ft. |