1. A right circular cylinder and a right circular cone have equal base radii and equal heights. The ratio of their volumes = ___________.

2. A solid hemisphere and a solid cone have equal base diameters and equal heights. Calculate and write the ratio of their volumes and the ratio of their curved surface areas.

3. A right circular cylinder and a right circular cone have equal base radii and equal heights. The ratio of their volumes is _________.

4. If a hemisphere and a cone have equal base areas and equal heights, then what will be the ratio of their volumes?

(a) 1:3 (b) 2:3 (c) 3:2 (d) 3:1

5. A right circular cylinder and a cone have equal bases, and the ratio of their volumes is 3:2. Prove that the height of the cone is half the height of the cylinder.

6. If a solid hemisphere and a solid cone have equal base diameters and equal heights, find the ratio of their curved surface areas.

7. A solid hemisphere and a solid right circular cylinder have the same height and the same base radius. Determine the ratio of their curved surface areas.

8. If a solid cone and a solid right circular cylinder have the same base radius and same height, and the ratio of their curved surface areas is 5:8, then determine the ratio of their base radius to height.

9. "A right circular cylinder and a right circular cone have a base radius ratio of 3:4 and a height ratio of 2:3; write the ratio of their volumes."

10. If a solid hemisphere and a solid cone have equal base areas and equal heights, then what is the ratio of their volumes?

(a) 1:3 (b) 1:2 (c) 2:3 (d) 3:4

11. If a right circular cone and a hemisphere have equal volumes and equal bases, then their height ratio will be 2:1.

12. A hemisphere and a cone have the same base and height. Determine the ratio of their volumes and the ratio of their curved surface areas.

13. If a hemisphere and a cone have the same base area and the same height, find the ratio of their curved surface areas.

14. A cone and a sphere have equal volumes. If their radii are the same, find the ratio of the height of the cone to the diameter of its base.

15. A solid hemisphere and a solid cone have the same base radius and height. Determine the ratio of their volumes.

(a) 3:2 (b) 11:3 (c) 2:3 (d) 4:3

16. A solid hemisphere and a solid cone have the same base diameter and height. Determine the ratio of their volumes and the ratio of their curved surface areas.

17. A solid object has its lower part in the shape of a hemisphere and its upper part in the shape of a right circular cone. If the surface areas of both parts are equal, determine the ratio of the radius to the height of the cone.

18. Determine the volume ratio of a solid cone, a solid hemisphere, and a solid cylinder that have equal base diameters and equal heights.

19. A solid right circular cone and a solid sphere have equal radii and equal volumes. Calculate and write the ratio of the diameter of the sphere to the height of the cone.

20. The bottom part of a solid is in the shape of a hemisphere, and the top part is in the shape of a right circular cone. If the surface areas of both parts are equal, then calculate and write the ratio of the radius to the height of the cone.

21. If two solid right circular cones have equal volume and their heights are in the ratio 1:2, then what is the ratio of the lengths of their radii?

(a) \(1:\sqrt2\) (b) \(1:2\) (c) \(\sqrt2 : 1\) (d) \(2:1\)

22. The ratio of the base radii of two right circular cones is 2 : 3, and the ratio of their heights is 3 : 2. Then the ratio of their volumes is —

(a) 3:2 (b) 9:4 (c) 4:9 (d) 2:3

23. Two right circular cones of equal height have their base diameters in the ratio 2 : 3. The ratio of their volumes will be —

(a) 3:4 (b) 4:5 (c) 5:4 (d) 4:9

24. A right circular cone has equal height and diameter. Determine the ratio of the curved surface area to the base area of the cone.

25. A right circular cone and a cylinder have equal curved surface areas. If the height and radius of the cone are \(h\) and \(r\), and the height and radius of the cylinder are \(H\) and \(r\), then show that: \[ h^2 = (2H + r)(2H - r) \]

26. A solid hemisphere and a solid cone have the same base radius and height. What is the ratio of their volumes?

(a) 3:2 (b) 1:3 (c) 2:3 (d) None of the above

27. A solid sphere and a solid cone have the same diameter, and their total surface areas are equal. What is the ratio of the cone's height to its diameter?

28. A right circular cone and a right circular cylinder with the same base and the same height will have a volume ratio of 1 : 3.

29. If two cones have equal base radii and their heights are in the ratio 2:3, then the ratio of their volumes will be ______.

30. If a solid hemisphere and a solid cone have the same base radius and height, what is the ratio of their volumes?

(a) 3:2 (b) 1:3 (c) 2:3 (d) None of the above