1. Two circular holes, each with a radius of 0.7 cm, are made on the curved surface of a hemisphere with a radius of 14 cm. Find the area of the metallic sheet of the hemispherical part.

2. The ratio of the total surface area of a sphere and a hemisphere with a unit radius will be 4:3.

3. What is the volume of the largest cube that can be cut out from a sphere with radius \(4\sqrt{3}\) cm?

(a) 512 cubic cm (b) 64 cubic cm (c) 216 cubic cm (d) 729 cubic cm

4. A solid sphere has a surface area of 616 square cm. It is melted to form 14 identical right circular cones, each with a height of 2 cm. Find the diameter of the base of each cone.

5. The lateral surface area of a right circular cone with a base radius of 1.5 meters and a slant height of 14 meters is _____.

(a) 66 square meters (b) 22 square meters (c) 44 square meters (d) 88 square meters

6. The lateral surface area of a right circular cone with a base radius of 1.5 meters and a slant height of 14 meters is _____.

(a) 66 square meters (b) 22 square meters (c) 44 square meters (d) 88 square meters

7. A sphere with a radius of 14 cm has two circular holes on its curved surface, each with a radius of 0.7 cm. Find the area of the remaining metallic sheet on the spherical surface.

8. A globe has a radius of 14 cm. Two circular holes, each with a radius of 0.7 cm, are made on the curved surface along the globe’s axis. Calculate the area of the metallic sheet covering the spherical part of the globe.

9. Two identical solid hemispheres, each with a base radius of \(r\) units, are joined together along their flat surfaces. The total surface area of the resulting solid body will be \(6πr^3\) square units.

10. Find the total surface area of a solid hemisphere with radius \(x\) units.

11. A solid sphere has a total surface area of 1386 cm². It is melted and reshaped into several right circular cones, each with a radius of 3.5 cm and a height of 6 cm. Find the radius of the solid sphere and determine how many such cones are formed.

12. 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?

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 the base radius of a right circular cone is 3 cm and the height is 4 cm, then the lateral surface area of the cone will be.

(a) \(10\pi \, cm^2\) (b) \(15\pi \, cm^2\) (c) \(12\pi \, cm^2\) (d) \(18\pi \, cm^2\)

15. The volume ratio of two solid spheres is 4:27. The ratio of their curved surface areas is-

(a) 4:9 (b) 2:3 (c) 16:35 (d) 8:27

16. 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

17. If the ratio of the volume of a sphere with radius ???? to the volume of a cone with the same radius is 4:9, then the height of the cone will be..

(a) \(3r\) (b) \(4r\) (c) \(6r\) (d) \(9r\)

18. The diameter of one sphere is twice the diameter of another sphere. The curved surface area of the first sphere is equal to the volume of the second sphere. What is the radius of the first sphere?

(a) 21 (b) 22 (c) 23 (d) 24

19. 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

20. 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

21. If the total surface area of a solid sphere is 462 square centimeters, what is its diameter?

(a) 7 cm (b) 14 cm (c) 21 cm (d) 3.5 cm

22. 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

23. If the radius of a sphere is increased by 2 cm, its curved surface area increases by 352 square centimeters. What was the original radius of the sphere?

(a) 5 cm (b) 6 cm (c) 7 cm (d) 5.6 cm

24. If the radius of a sphere is increased by 3%, by what percentage will its surface area increase?

(a) 6.09% (b) 7% (c) 5.06% (d) 9%

25. A solid lead sphere with a diameter of 12 cm is melted to form three smaller spheres. If the diameters of the smaller spheres are in the ratio 3:4:5, what is the radius of the smallest sphere?

(a) 1.5 cm (b) 3 cm (c) 4 cm (d) 5 cm

26. 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

27. If the radius of a cone is \(r\) units, the height is \(h\) units, and the curved surface area is \(S\) square units, then prove that \[ h = \frac{\sqrt{S^2 - \pi^2 r^4}}{\pi r} \]

28. Two solid spheres with radii of 1 cm and 6 cm are melted and recast into a hollow sphere with a thickness of 1 cm. Find the outer radius of the new hollow sphere and calculate its curved surface area.

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

30. A hollow vertical cylindrical iron pipe has an outer radius of 5 cm and an inner radius of 4 cm. If the total surface area of the pipe is 1188 square cm, what is the length of the pipe?