1. The numerical value of the volume and the total surface area of a solid hemisphere are equal. Write the radius of the hemisphere.

2. The total surface area and volume of a solid hemisphere are numerically equal. Find the radius of the base of the hemisphere.

3. The volume and total surface area of a solid hemisphere are numerically equal. Then the diameter of the hemisphere will be –

(a) 4:5 units (b) 6 units (c) 9 units (d) 3 units

4. The numerical value of a sphere’s volume and surface area are equal. What is the numerical value of the sphere’s radius?

(a) 2 (b) 3 (c) 4 (d) 5

5. The total surface area of a solid hemisphere and the curved surface area of a solid sphere are equal. What is the ratio of the radius of the hemisphere to the radius of the sphere?

6. The numerical value of the volume and the lateral surface area of a right circular cone are equal. If the height and radius of the cone are \( h \) units and \( r \) units respectively, find the value of \[ \left( \frac{1}{h^2} + \frac{1}{r^2} \right) \]

7. 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 \(\frac{1}{h^2} + \frac{1}{r^2}\).

8. If the radius of a right circular cylinder is 2 units, then for any height, the numerical values of the cylinder’s volume and curved surface area will be equal.

9. The numerical value of the volume and the lateral surface area of a right circular cone are equal. If the height of the cone is \( h \) and the radius is \( r \), determine the value of \(\cfrac{1}{h^2}+\cfrac{1}{r^2}\).

10. If the radius of a right circular cylinder is 2 units, then for any height of the cylinder, the numerical values of its volume and curved surface area are equal.

11. If the radius of a right circular cylinder is 2 units, then for any height of the cylinder, the numerical values of its volume and curved surface area are equal.

12. If the radius of a right circular cylinder is 2 units, then for any height, the numerical values of the cylinder’s volume and its curved surface area will be equal.

13. If the numerical value of the volume and the surface area of a sphere are equal, what is the numerical value of the radius of the sphere?

(a) 3 (b) 5 (c) 7 (d) 4

14. 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

15. If the numerical values of the curved surface area and the total surface area of a sphere are equal, what is its radius?

(a) 3 units (b) 4 units (c) 5 units (d) 6 units

16. If the numerical values of the volume and curved surface area of a sphere are equal, then its radius is _____ units.

17. If the total surface area and the volume of a cube have the same numerical value, determine the length of its diagonal.

18. If the numerical values of the volume and lateral surface area of a cone are equal, then the relationship between its height and base radius is?

(a) \(r^2-h^2=2^2\) (b) \(2r^2+h^2=3^2\) (c) \(r^2+h^2=3^2\) (d) none of the above

19. The curved surface area and the total surface area of a solid hemisphere are equal.

20. If the curved surface area of a solid hemisphere is \( S \) and its volume is \( V \), find the value of \( \cfrac{S^3}{V^2} \).

21. If the numerical values of the volume and surface area of a solid sphere are equal, then what is the numerical value of its diameter?

22. If the numerical value of the curved surface area and the volume of a sphere are equal, then the radius will be 3 units.

23. A solid right circular cylinder and two hemispheres have equal base radii. If the two hemispheres are attached to the flat circular ends of the cylinder, then the total surface area of the resulting solid = curved surface area of one hemisphere + curved surface area of the cylinder + curved surface area of the other hemisphere.

24. "When a right circular cone has equal numerical values for its volume and lateral surface area, and its height and radius are \(h\) units and \(r\) units respectively, write the value of \(\frac{1}{h^2} +\frac{1}{r^2}\)."

25. The ratio of the total surface areas of a solid sphere and a solid hemisphere of the same radius is 2:1.

26. If the ratio of the volume to the total surface area of a solid hemisphere is 4:9, what is the radius of the hemisphere?

27. If a sphere's volume and surface area have equal numerical values, then the numerical value of its radius is ——.

28. In a right-angled triangle, the lengths of the two sides adjacent to the right angle are 4 cm and 3 cm respectively. If the triangle is rotated once completely around the longer of the two adjacent sides, find the total surface area and the volume of the solid formed.

29. If the numerical value of the volume and the curved surface area of a right circular cone are equal, find the length of the diameter of the cone — in units.

30. 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?