31. The radius of the base of a cone is 4 inches: find the height, so that the volume may be equal to that of a sphere with diameter 4 inches.

32. The height of a cone is 12 inches: find the radius of its base, so that the volume may be equivalent to that of a sphere with diameter 6 inches.

33. The circumference of the base of a cone is 32 feet: find the height so that the volume may be equivalent to that of a sphere with diameter 10 feet.

34. A solid is composed of a hemisphere and a cone on opposite sides of the same circular base; the diameter of this base is 5 feet, and the height of the cone is 5 feet: find the volume of the solid.

35. Find how many times larger the Earth is than the Moon, taking the diameter of the Earth as 7900 miles, and that of the Moon as 2160 miles.

The following examples involve the extraction of the cube root:

36. Find the length of a cube which shall be equivalent in volume to a sphere 20 inches in diameter.

37. Find the diameter of a sphere which shall be equivalent in volume to a cube 20 inches in length.

38. Find the diameter of a sphere which shall be equivalent in volume to a cylinder, the radius of the base of which is 8 inches and the height 12 inches.

39. If 30 cubic inches of gunpowder weigh one lb., find the internal diameter of a hollow sphere which will hold 15 lbs.

40. If a leaden ball of one inch diameter weigh lb., find the diameter of a leaden ball which weighs 588 lbs.

41. If a cubic inch of metal weigh 6:57 ounces, find the diameter of a sphere of the metal which weighs 220-16

ounces.

42. A Stilton cheese is in the form of a cylinder, and a Dutch cheese in the form of a sphere. Determine the diameter of a Dutch cheese which weighs 9 lbs., when a Stilton cheese, 14 inches high and 8 inches in diameter, weighs 12 lbs.

XXX. ZONE AND SEGMENT OF A SPHERE.

298. To find the volume of a zone of a sphere.

RULE. To three times the sum of the squares of the radii of the two ends, add the square of the height; multiply the sum by the height, and the product by 5236: the result will be the volume.

299. Examples.

(1) The radii of the ends are 8 inches and 11 inches, and the height is 2 inches.

64+121=185; 3 × 185=555; 555+4=559;

559 × 2 × 5236=585 3848.

Thus the volume is about 585 cubic inches.

(2) The radius of each end is 20 inches, and the height is 9 inches.

400+400-800; 3 × 800-2400, 2400+81=2481;

2481 × 9 × 5236=11691·4644.

Thus the volume is nearly 11691.5 cubic inches.

300. The Rule given in Art. 298 will serve to find the volume of a segment of a sphere, if we remember that the radius of one end of a segment is nothing; but it may be convenient to state the Rule for this case explicitly.

301. To find the volume of a segment of a sphere.

RULE. To three times the square of the radius of the base add the square of the height; multiply the sum by the height, and the product by 5236: the result will be the volume.

302. Examples.

(1) The radius of the base is 5 inches, and the height 3 inches.

3 × 25=75; 75+9=84; 84 × 3 × 5236 = 131.9472.

Thus the volume is nearly 132 cubic inches.

(2) The diameter of the base of a segment is 3 feet, and the height is 9 inches.

303. We will now solve some exercises.

(1) The height of a segment of a sphere is 3 inches, and the diameter of the sphere is 14 inches: find the volume of the segment.

We must first determine the square of the radius of the base of the segment. Using the diagram of Art. 78 we have ED=3 inches, and EF-14 inches. By Art. 89 we shall find that the square of AD=33.

33 x3=99; 99+9=108; 108 × 3 × 5236=169.6464. Thus the volume is about 170 cubic inches.

(2) The radius of the base of a segment is 24 inches, and the radius of the sphere is 25 inches: find the volume.

We must first determine the height of the segment. Using the diagram of Art. 78, we have AD=24 inches, and AC 25 inches. By Art. 60 we shall find that CD=7 inches. Therefore DE=18 inches. The square of 24= 576; 576 × 3 = 1728; the square of 18=324;

1728+324=2052.

2052 × 18 x 5236 = 19339'6896. Thus the volume is nearly 19340 cubic inches.

EXAMPLES. XXX.

1. The radii of the ends of a zone of a sphere are 7 inches and 8 inches; and the height is 3 inches: find the volume.

2. The radii of the ends of a zone of a sphere are 8 inches and 12 inches; and the height is 6 inches: find the volume.

3. The height of a segment of a sphere is 6 feet and the diameter of the base is 8 feet: find the volume.

4. The height of a segment of a sphere is 2 feet 8 inches and the diameter of the base is 8 feet: find the volume.

5. The height of a segment of a sphere is 4 feet, and the diameter of the sphere is 12 feet: find the volume.

6. The height of a segment of a sphere is 5 feet and the diameter of the sphere is 15 feet: find the volume.

7. The radius of the base of a segment of a sphere is 12 feet and the radius of the sphere is 13 feet: find the volume of the segment.

8. The radius of the base of a segment of a sphere is 8 feet and the radius of the sphere is 17 feet: find the volume of the segment.

9. The diameter of a sphere is 20 feet: find the volumes of the two segments into which the sphere is divided by a plane, the perpendicular distance of which from the centre is 5 feet.

10. The diameter of a sphere is 18 feet: the sphere is divided into two segments, one of which is twice as high as the other: find the volume of each.

11. The radius of the base of a segment of a sphere is 1 inch, and the radius of the sphere is 24 inches: find the volume of the segment.

12. Find the weight of a pair of iron dumb-bells, each consisting of two spheres of 4 inches diameter, joined by a cylindrical bar, 6 inches long and 2 inches in diameter; an iron ball 4 inches in diameter weighing 9 lbs.

13. The diameter of a sphere is 9 feet; the sphere is divided into three parts of equal height by two parallel planes: find the volume of each part.

14. A sphere, 16 inches in diameter, is divided into four parts of equal height by three parallel planes: find the volume of each part.

15. Find the volume of a zone of a sphere, supposing the ends to be on the same side of the centre of the sphere, and distant respectively 10 inches and 15 inches from the centre; and the radius of the sphere to be 20 inches.

16. Find the volume of a zone of a sphere supposing the ends to be on opposite sides of the centre of the sphere, and distant respectively 10 inches and 15 inches from the centre; and the radius of the sphere to be 20 inches.

17. A bowl is in the shape of a segment of a sphere; the depth of the bowl is 9 inches, and the diameter of the top of the bowl 3 feet: find to the nearest pint the quantity of water the bowl will hold.

18. Verify by calculating various cases the following statement: if the height of a segment of a sphere is threefourths of the radius of the sphere the volume of the segment is three fourths of the volume of a sphere which has its radius equal to the height of the segment.

T. M.

121