18. The height of a right cone is 10m, the radius of its base 5m. What must be the distance from the base of a plane parallel to the base in order that the volume of the frustum made by the plane may be 20cbm?

19. The frustum of a right cone is 14 ft. high, and has a volume of 924 cub. ft. Find the radii of its bases if their sum is 9 ft.

20. From a right cone whose slant height is 30 ft., and circumference of whose base is 10 ft., there is cut off by a plane parallel to the base a cone whose slant height is 6 ft. Find the convex surface and the volume of the frustum.

21. The interior dimensions of an iron vessel, open at the top and in the shape of the frustum of a right cone, are: diameter at the top 1 ft. 9 in., at the bottom 7 in., depth 1 ft. 8 in.; and the corresponding exterior dimensions are 2 ft., 9 in., and 1 ft. 10 in. How many cubic inches of iron were used in its construction?

22. A Dutch windmill in the shape of the frustum of a right cone is 12m high. The outer diameters at the bottom and the top are 16m and 12m, the inner diameters 12m and 10m, respectively. How many cubic meters of stone were required to build it?

23. The chimney of a factory has the shape of the frustum of a regular pyramid. Its height is 180 ft., and its upper and lower bases are squares whose sides are 16 ft. and 10 ft., respectively. The section of the flue is throughout a square whose side is 7 ft. How many cubic feet of material does the chimney contain?

24. Find the volume V of the frustum of a cone of revolution, having given the slant height a, the height h, and the convex surface S.

$7. THE SPHERE.

1. What is the locus of the tangents which can be drawn from an exterior point to a given sphere?

2. The radius of a sphere is 2m. Find the area of a section made by a plane, the distance of which from the centre of the sphere is 40cm.

3. What is the distance of a small circle the area of which is 3am, from the centre of a sphere the radius of which is 2m?

4. The radius of a sphere is 7 ft. Find the distance from the pole of a great circle to its circumference, (i.) when measured on the surface of the sphere; (ii.) when measured in a straight line.

5. The distance from the two poles of a small circle drawn on the surface of a sphere to the circumference of the circle are 3 and 4m respectively. Find the area of this circle.

6. In order to find the radius of a sphere, describe a small circle with a pair of dividers, and measure the length of its circumference with the help of a string. If the opening of the dividers is 6 in., and the length of the circumference is 22 in., find the radius of the sphere.

7. Find the surface of a sphere if the diameter is (i.) 10 in.; (ii.) 1 ft. 9 in.; (iii.) 2 ft. 4 in.; (iv.) 7 ft.; (v.) 4.2 ft.; (vi.) 10.5 ft.

8. Find the diameter of a sphere if the surface is (i.) 616 sq. in.; (ii.) 38 sq. ft.; (iii.) 9856 sq. ft.

9. The circumference of a dome in the shape of a hemisphere is 66 ft.; how many square feet of lead are required to cover it?

10. How many square feet of lead are required to make 16 hemispherical bowls, if the diameter of each is 2 ft. 4 in. ?

11. If the ball on the top of St. Paul's Cathedral in London is 6 ft. in diameter, what would it cost to gild it at 7 cents per square inch?

12. If the surface of a sphere is S, what is the surface of a sphere having a radius twice as large?

13. If the surface of a sphere is S, find the circumference C of a great circle.

14. What is the numerical value of the radius of a sphere if its surface has the same numerical value as the circumference of a great circle?

15. Find the surface of a lune if its angle is 30°, and the total surface of the sphere is 4 sq. ft.

16. What is the angle of a lune if its surface is 1 sq. ft., and the total surface of the sphere is 9 sq. ft.?

17. What fractional part of the whole surface of a sphere is a spherical triangle whose angles are 43° 27', 81° 57′, and 114° 36'?

18. The angles of a spherical triangle are 60°, 70°, and 80°. The radius of the sphere is 14 ft. Find the area of the triangle in square feet.

19. The sides of a spherical triangle 128°. The radius of the sphere is 14 ft. the polar triangle in square feet.

are 38°, 74°, and Find the area of

20. The sides of a spherical triangle are each equal to 90°. What part of the whole surface of the sphere does the triangle contain?

21. The radius of a sphere is 7 ft. Find the area of a spherical triangle formed by the arcs of three great circles, two of which are perpendicular to the third, and make with each other an angle of 60°.

22. Through a diameter of a sphere two planes are passed so as to cut from the circumference of a great circle, perpendicular to both the planes, an arc of 45°. The area of the spherical surface included by these planes is 16π sq. ft. Find the radius of the sphere.

23. On a sphere whose radius is 21 ft., find the area of a spherical triangle polar to one whose perimeter is equal to half the circumference of a great circle.

24. Find the area of a spherical polygon on a sphere whose radius is 10 ft., if its angles are 100°, 120°, 140°, and 160°.

25. The planes of the faces of a quadrangular spherical pyramid make with each other angles of 80°, 100°, 120°, and 150°; and the length of a lateral edge of the pyramid is 42 ft. Find the area of its base in square feet.

26. The planes of the faces of a triangular spherical pyramid make with each other angles of 40°, 60°, and 100°, and the area of the base of the pyramid is 4π sq. ft. Find the radius of the sphere.

27. The diameter of a sphere is 21 ft. Find the curved surface of a segment whose height is 5 ft.

28. What is the area of a zone of one base whose height is h, and the radius of the base r? What would be the area if the height were twice as great?

29. In a sphere whose radius is r, find the height of a zone whose area is equal to that of a great circle.

30. The radius of a sphere is 4 ft. The sphere is cut by two parallel planes situated on the same side of the centre, at the distances 2 ft. and 3 ft. respectively from the centre. Find the area of the zone formed by the planes, and the area of each of the circles which form its bases.

31. If the radius of a sphere is r, and the height of a zone on the sphere is h, find the radius of a circle equivalent to the area of the zone.

32. How many square feet of lead will be required to line the inside of a bowl in the shape of a segment of a sphere which measures 40 in. across the top, and whose greatest depth is 10 in. ?

33. Find the convex surface of a slice 2 ft. high, cut from a sphere whose radius is 17 ft.

34. The height of a spherical zone is 8 ft.; its bases are on opposite sides of the centre, and their radii are 10 ft. and 6 ft. Find the area of the zone.

35. The altitude of the torrid zone is about 3200 miles. Find its area in square miles, assuming the earth to be a sphere with a radius of 4000 miles.

36. A plane divides the surface of a sphere of radius r into two zones, such that the surface of the greater is a mean proportional between the entire surface and the surface of the smaller. Find the distance of the plane from the centre of the sphere.

37. If a sphere of radius r is cut by two planes equally distant from the centre, so that the area of the zone comprised between the planes is equal to the sum of the areas of its bases, find the distance of either plane from the centre.

38. Find the area of the zone generated by an arc of 30°, of which the radius is r, and which turns around a diameter passing through one of its extremities.