the base by Art. 171. Divide 8 by 3·1416; the quotient is 2.5464...; the square root of this is 1.5958 very nearly. Thus the radius of the base is 1·5958 feet very nearly. (2) The volume of a right circular cone is 20 cubic feet; the height is twice the radius of the base: find the area of the whole surface. By Art. 263 we see that the product of the cube of 2 the radius of the base into of 3.1416 is equal to 20; so 3 therefore the radius of the base in feet is the cube root of 9.549... this we shall find to be 2.1215.... Therefore the area of the base in square feet is the product of 31416 into the square of 2121: it will be found that this is 14.140... Now if the radius of the base were 1 foot, and the height of the cone were 2 feet, the slant height would be √5 feet, by Art. 55; that is, the slant height would be 5 times the radius of the base. And thus in the present case since the height is twice the radius of the base, the slant height is 5 times the radius of the base: that is the slant height is 2.236... times the radius of the base. Hence, by Art. 337, the area of the curved surface in square feet is the product of 2.236... into 14:140... that is 31617... Therefore the whole area of the surface in square feet =14140+31 617=45 757. (3) The volume of a right circular cone is 20 cubic feet; the slant height is three times the radius of the base : find the area of the whole surface. If the radius of the base were 1 foot, and the slant height 3 feet, the height would be 8 feet by Art. 60; that is, the height would be 8 times the radius of the base. And thus in the present case since the slant height is 3 times the radius of the base, the height is 8 times the radius of the base. Then, as in the preceding Exercise we see that the cube of the radius of the base = 3 x 20 √8×3.1416 ; it will be found that this is 6752...; therefore the radius of the base is the cube root of 6752...: it will be found that this is 1.890.... By Art. 337, the area of the whole surface is 4 times the area of the base; so that in square feet it is 4 times the product of 3.1416 into the square of 1'890...: it will be found that this is 44.888... The whole surface is less in this case than in the case of the preceding Exercise. In fact, it will be found by comparing the results of Examples 31...40 at the end of the present Chapter, and of similar Examples, that if the whole surface of a right circular cone be given the volume is greatest when the slant height is three times the radius; and if the volume of a right circular cone be given, the whole surface is least when the slant height is three times the radius. EXAMPLES. XXXVI. Find in square inches the area of the curved surface of right circular cones having the following dimensions: 1. Slant height 2 feet 3 inches, circumference of base 4 feet 5 inches. 2. Slant height 3 feet 2 inches, circumference of base 5 feet 7 inches. 3. Slant height 2 feet, radius of base 1 foot 9 inches. 4. Slant height 2 feet 8 inches, radius of base 2 feet 10 inches. 5. Slant height 3 feet, radius of base 1 foot 6 inches. 6. Height 2 feet, radius of base 7 inches. 7. Height 3 feet 4 inches, radius of base 9 inches 8. Height 2 feet 6 inches, radius of base 1 foot 4 inches. 9. Height 5 feet, radius of base 11 inches. 10. Height 4 feet 8 inches, radius of base 2 feet 9 inches. 11. Height 5 feet, perimeter of base 6.2832 feet. Find in square feet the area of the whole surface of right circular cones having the following dimensions: 13. Slant height 4 feet, radius of base 2 feet. 14. Slant height 5.3 feet, radius of base 3.2 feet. 18. Height 1 foot 9 inches, radius of base 1 foot 8 inches. 19. Height 18 inches, circumference of base 27 inches. 20. Height 4 feet, circumference of base 7 feet. 21. The area of the curved surface of a right circular cone is 750 square inches, and the circumference of the base is 50 inches: find the slant height. 22. The area of the curved surface of a right circular cone is 800 square inches, and the circumference of the base is 64 inches: find the height of the cone. 23. The area of the curved surface of a right cone is 12 square feet, and the radius of the base is 15 feet: find the slant height. 24. The area of the curved surface of a right cone is 25 square feet, and the radius of the base is 2·25 feet: find the height of the cone. 25. The area of the curved surface of a right cone is 650 square inches, and the slant height is 25 inches: find the circumference of the base. 26. The area of the curved surface of a right circular cone is 18 square feet, and the slant height is 33 feet: find the radius of the base. 27. The area of the whole surface of a right circular cone is 15 square feet, and the slant height is three times the radius of the base: find the radius of the base. 28. The area of the whole surface of a right circular cone is 19 square feet, and the slant height is four times the radius of the base: find the radius of the base. 29. Find what length of canvass three quarters of a yard wide is required to make a conical tent 12 feet in diameter and 8 feet high. 30. Find what length of canvass two-thirds of a yard wide is required to make a conical tent 16 yards in diameter and 10 feet high. The area of the whole surface of a right circular cone is 100 square feet; find in cubic feet the volume in the following cases: 31. The slant height twice the radius of the base. 32. The slant height three times the radius of the base. 33. The slant height four times the radius of the base. 34. The slant height five times the radius of the base. 35. The slant height six times the radius of the base. The following examples involve the extraction of the cube root. The volume of a right circular cone is 31416 cubic inches; find in square inches the whole surface in the following cases: 36. Height equal to the radius of the base. 37. Height equal to twice the radius of the base. 38. Height equal to three times the radius of the base. 39. Height equal to half the radius of the base. 40. Height equal to a third of the radius of the base. XXXVII. FRUSTUM OF A RIGHT CIRCULAR CONE. 339. The surface of a frustum of a right circular cone consists of two circular ends and another portion which we shall call the curved surface. 340. Let ABCD be a sector of a circle. With A as centre and any radius less than AB describe the arc EFG. Let the piece BCDGFE be cut out of paper or cardboard. Then let it be bent round until the edge EB just comes into contact with the edge GD. It is easy to see that by proper adjustment we can thus obtain a thin shell the B E F outside of which will correspond to the curved surface of a frustum of a right circular cone; EFG becomes the circumference of one end of the frustum, and BCD becomes the circumference of the other end; EB becomes the slant height of the frustum. Hence it will follow that the curved surface of a frustum of a right circular cone is equal to the difference of two sectors of circles which have a common angle; the arcs of the sectors being the circumferences of the ends of the frustum, and the difference of their radii being the slant height of the frustum: thus from the last Rule of Art. 183 we obtain the Rule which will now be given. 341. To find the area of the curved surface of a frustum of a right circular cone. RULE. Multiply the sum of the circumferences of the two ends of the frustum by the slant height of the frustum, and half the product will be the area of the curved surface. |