1. If a right circular cone and a sphere have the same radius, and the numerical value of the cone's volume is equal to the numerical value of the sphere's curved surface area, then what is the height of the cone?

(a) 10 units (b) 11 units (c) 12 units (d) 13 units

2. A solid right circular cone and a solid sphere have the same radius and the same volume. What is the ratio of the diameter of the sphere to the height of the cone?

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

3. Two solid spheres with radii of 8 cm and 10 cm are melted to form a right circular cone with a height of 42 cm. What is the radius of the cone?

(a) 8 cm (b) 10 cm (c) 12 cm (d) 16 cm

4. 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: \[ h^2 = (2H + r)(2H - r) \]

5. A right circular cone and a hemisphere have equal base areas and volumes. Determine the ratio of their heights.

6. A sphere and a right circular cylinder both have a radius of 3 cm. If their volumes are equal, what is the height of the cylinder?

7. The radius of a right circular cylinder is \(r\) units, and its height \(R\) units is \(\frac{2}{3}\) times the diameter of a sphere of radius \(R\). If their volumes are equal, show that \(r = R\).

8. 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.

9. If a solid sphere and a solid right circular cylinder have the same radius and equal volume, then write the ratio of the cylinder's radius to its height.

10. 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.

11. 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.

12. If a solid sphere and a solid right circular cylinder have equal radii and the same volume, then calculate and write the ratio of the radius to the height of the cylinder.

13. A solid sphere with a radius of \(r\) units is melted and recast into a solid right circular cone with a height of \(r\) units. Find the radius of the base of the cone.

(a) 2r units (b) 3r units (c) r unit (d) 4r units

14. The base radius of a solid right circular cone is \(r\) units, its vertical height is \(h\) units, and its slant height is \(l\) units. The cone’s base is attached to the base of a solid right circular cylinder. If the base radius and height of the cylinder are equal to those of the cone, then the total surface area of the combined solid is \((πrl + 2πrh + 2πr^2)\) square units.

15. 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.

16. If the height of a right circular cone remains unchanged and its radius is increased by 1 unit, the volume of the cone becomes four times its original volume. What is the length of the diameter of the cone?

17. The radius of the base of a right circular cone is equal to the radius of a sphere. The volume of the cone (in cubic centimeters) is equal to the curved surface area of the sphere (in square centimeters). What is the height of the cone?

(a) 4 cm (b) 3 cm (c) 8 cm (d) 12 cm

18. What is the ratio of the volumes of a right circular cylinder and a cone having the same radius and height?

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

19. What is the ratio of the volumes of a solid right circular cylinder, a solid right circular cone, and a solid sphere, all having the same diameter and the same height?

20. If a right circular cone has a height of 12 cm and a volume of \(100\pi\) cubic cm, then what is the radius of the cone?

21. 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

22. 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?

23. If the radius of the base of a right circular cone is halved and its height is doubled, the volume of the cone remains the same.

24. 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.

25. If the radius of a right circular cone is decreased by 50% and its height is increased by 50%, what is the percentage change in the volume of the cone?

26. 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

27. The volume and the lateral surface area of a right circular cone are numerically equal. If the height and radius of the cone are \(h\) and \(r\) units respectively, write the value of \(\frac{1}{h^2} + \frac{1}{r^2}\).

28. If the radius of a right circular cone is halved and its height is doubled, then the volume of the cone will be — of its original volume.

29. The radius of a hemisphere and the height of a cone are equal. If their volumes are numerically equal, then what is the ratio of the radius of the hemisphere to the radius of the cone?

30. 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