19. Volume 60.7 cubic feet; height 5'45 feet. 21. The faces of a pyramid on a square base are equilateral triangles, a side of the base being 120 feet: find the volume. 22. Find the volume of a pyramid which stands on a square base, each side of which is 200 feet, each of the edges which meet at the vertex being 150 feet. 23. A pyramid has a square base, the area of which is 20 25 square feet; each of the edges of the pyramid which meet at the vertex is 30 feet: find the volume. 24. The base of a pyramid is a rectangle 80 feet by 60 feet; each of the edges which meet at the vertex is 130 feet: find the volume. 25. The base of a pyramid is a square, each side of which is 24 feet; the length of the straight line drawn from the vertex to the middle point of any side of the base is 218 feet: find the volume. 26. The base of a pyramid is a square, each side of which is 12 feet; the length of the straight line drawn from the vertex to the middle point of any side of the base is 25 feet: find the volume. 27. The base of a pyramid is a rectangle, which is 21 feet by 25; the length of the straight line drawn from the vertex to the middle point of either of the longer sides of the base is 23.3 feet: find the volume. 28. The base of a pyramid is a rectangle, which is 18 feet by 26; the length of the straight line drawn from the vertex to the middle point of either of the shorter sides of the base is 24 feet: find the volume. 29. The slant side of a right circular cone is 25 feet, and the radius of the base is 7 feet: find the volume. 30. The section of a right circular cone by a plane through its vertex perpendicular to the base is an equilateral triangle, each side of which is 12 feet: find the volume of the cone. 31. The slant side of a right circular cone is 41 feet, and the height is 40 feet: find the volume. 32. The slant side of a right circular cone is 55 feet, and the height is 42 feet: find the volume. 33. Find how many gallons are contained in a vessel which is in the form of a right circular cone, the radius of the base being 8 feet, and the slant side 17 feet. 34. A conical wine glass is 2 inches wide at the top and 3 inches deep: find how many cubic inches of wine it will hold. 35. Find the volume of a circular cone, the height of which is 15 feet, and the circumference of the base 16 feet. 36. A cone, 3 feet high and 2 feet in diameter at the bottom, is placed on the ground, and sand is poured over it until a conical heap is formed 5 feet high and 30 feet in circumference at the bottom: find how many cubic feet of sand there are. 37. The volume of a cone is 22 cubic feet; the circumference of the base is 9 feet: find the height. 38. Find the number of cubic feet in a regular hexagonal room, each side of which is 20 feet in length, and the walls 30 feet high, and which is finished above with a roof in the form of a hexagonal pyramid 15 feet high. 39. Find the volume of the pyramid formed by cutting off a corner of the cube, whose side is 20 feet, by a plane which bisects its three conterminous edges. 40. The edge of a cube is 14 inches; one of the corners of the cube is cut off, so that the part cut off forms a pyramid, with each of its edges terminating in the angle of the cube, 6 inches in length: find the volume of the solid that remains. 41. The great pyramid of Egypt was 481 feet in height, when complete; and its base was a square 764 feet in length: find the volume to the nearest number of cubic yards. 42. The spire of a church is a right pyramid on a regular hexagonal base; each side of the base is 10 feet and the height is 50 feet; there is a hollow part which is also a right pyramid on a regular hexagonal base, the height of the hollow part is 45 feet, and each side of the base is 9 feet: find the number of cubic feet of stone in the spire. XXVI. FRUSTUM OF A PYRAMID OR CONE. 268. To find the volume of a frustum of a Pyramid or Cone. RULE. To the areas of the two ends of the frustum add the square root of their product; multiply the sum by the height of the frustum, and one-third of the product will be the volume. 269. Examples. (1) The area of one end of a frustum of a pyramid is 18 square inches, and the area of the other end is 98 square inches; and the height of the frustum is 15 inches. The square root of 18 × 98 is 42; 18+98 +42=158. 1 x 15 x 158=790. Thus the volume is 790 cubic inches. (2) The radius of one end of a frustum of a cone is 5 feet, and the radius of the other end is 3 feet; and the height of the frustum is 8 feet. = The area of one end in square feet 25 x 3'1416; the area of the other end in square feet 9 × 31416; the square root of the product of these numbers is 3:1416 multiplied by the square root of 9 × 25, that is, 3·1416 × 15. Add these results and we obtain 31416 multiplied by the sum of 25, 9, and 15; that is, 3·1416 × 49. Thus the volume is 4105024 cubic inches. 270. It will be seen that in the second example of the preceding Article we have adopted a peculiar arrangement, with the view of saving labour in multiplication. The same arrangement may always be used with advantage in finding the volume of the frustum of a cone, when the radii of the ends are known. In fact, in such a case we may, instead of the rule in Art. 268, adopt the following, which is sub stantially the same, but more convenient in form: add the squares of the radii of the ends to the product of the radii: multiply the sum by the height, and this product by 31416: one third of the result will be the volume. 271. We will now solve some exercises. (1) The radii of the ends of a frustum of a right circular cone are 7 inches and 10 inches respectively; and the slant height of the frustum is 5 inches. We must determine the height of the frustum. Let the diagram represent a section of the frustum made by a plane, containing the axis of the cone. We see that the slant height is the hypotenuse of a right-angled triangle, of which one side is the height of the frustum, and the other side is the difference of the radii of the ends. In the present case the slant height is 5 inches, and the difference of the radii of the ends is 3 inches; therefore, by Art. 60, the height of the frustum is 4 inches. Now 7x7=49, 10 × 10=100, 7×10=70; 1 49+100+70=219; 3 × 4 × 219 × 31416=917.3472. Thus the volume is 917.3472 cubic inches. (2) The ends of a frustum of a pyramid are equilateral triangles, the sides being 3 feet and 4 feet respectively; and the height is 9 feet. By Art. 206 the area of one end in square feet = 9 × 433... ; and the area of the other end in square feet 16 × 433... ; the square root of the product of these numbers = 12 × 433... Add these three results, and we obtain 37 × 433........ Thus the volume is rather more than 48 cubic feet. EXAMPLES. XXVI. Find in cubic feet the volumes of frustums of pyramids which have the following dimensions : 1. Areas of ends 45 square feet and 12.5 square feet; height 1.5 feet. 2. Areas of ends 4 square feet and 5 square feet; height 2 feet 6 inches. 3. Areas of ends 900 square inches and 6'5 square feet; height 2 yards. 4. Areas of ends 7.5 square feet and 8.25 square feet; height 6.125 feet. Find in cubic feet the volumes of frustums of cones which have the following dimensions: 5. Radii of the ends 3 feet and 4 feet; height 5 feet. 6. Radii of the ends 45 feet and 54 feet; height 6.5 feet. 7. Radii of the ends 48 feet and 64 feet; height 7.2 feet. 8. Radii of the ends 6.375 feet and 5'1 feet; height 10 feet. 9. The slant side of the frustum of a right circular cone is 5 feet, and the radii of the ends are 7 feet and 10 feet: find the volume. 10. Find the cost of the frustum of a right circular cone of marble at 24 shillings per cubic foot, the diameter of the greater end being 4 feet, of the smaller end 1 feet, and the length of the slant side 8 feet. |