1. A rectangular sheet of paper has a length of \(l\) units and a breadth of \(b\) units. The sheet is rolled to form a right circular cylinder, where the circumference equals the length of the paper. The curved surface area of the cylinder is ____ square units.

2. The length of a rectangular sheet of paper is \(l\) units and its breadth is \(b\) units. The paper is rolled to form a right circular cylinder such that the circumference of the base is equal to the length of the paper. The curved surface area of the cylinder is ____________ square units.

3. A rectangular sheet of paper has length \(l\) units and width \(b\) units. When rolled along its length, it forms a cylindrical tube whose circumference is equal to the length of the paper. The curved surface area of the cylinder is ___ square units.

4. 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.

5. Translate: "If a right circular cylinder has lateral surface area \(c\) square units, base radius \(r\) units, and volume \(V\) cubic units, then find the height (value) of the cylinder." Would you like me to solve it too? I’m ready when you are.

6. A hollow metallic sphere has an inner radius of 3 cm and an outer radius of 5 cm. It is melted to form a solid right circular cylinder of height \(\frac{8}{3}\) cm. Find the diameter of the base of the cylinder.

7. A right circular cylinder has a height-to-base-radius ratio of 3:1. If the volume of the cylinder is \(1029\pi\) cubic cm, find the total surface area of the cylinder.

8. If the total surface area of a cube is \(s\) square units and the length of its diagonal is \(d\) units, then the relationship between \(s\) and \(d\) will be –

(a) \(s=6d^2\) (b) \(3s=7d\) (c) \(s^3=d^2\) (d) \(d^2 = \cfrac{s}{2}\)

9. The curved surface area of a right circular cone is \(\sqrt{5}\) times its base area. The ratio of its height to radius will be _____.

10. If the curved surface area of a right circular cylinder is \(c\) square units, the base radius is \(r\) units, and the volume is \(v\) cubic units, then the value of \(\frac{cr}{v}\) is = _____.

11. If a right circular cylinder with radius \(2x\) cm and height \(x\) cm is melted to form a sphere, then the radius of the resulting sphere will be –

(a) \(\sqrt[3]3{x}\) cm (b) \(\sqrt[3]{3x}\) (c) \(x\) (d) \(2x\)

12. If the curved surface area of a right circular cylinder is \(c\) square units, the radius of the base is \(r\) units, and the volume is \(v\) cubic units, then find the value of \(\frac{cr}{v}\).

13. If the total surface area of a cube is \(s\) square units and the length of its diagonal is \(d\) units, then the relationship between \(s\) and \(d\) is \[ s = 6d^2 \]

14. The total surface area of an open-ended right circular cylinder is _____, where the base radius of the cylinder is \(r\).

15. If the curved surface area of a right circular cylinder is \( C \) square units, the base radius is \( r \) units, and the volume is \( V \) cubic units, determine the value of \( \cfrac{Cr}{V} \).

16. If the total surface area of a cube is \(s\) square units and the length of its diagonal is \(d\) units, then the relationship between \(s\) and \(d\) is \(s^3 = d^2\).

17. If the curved surface area of a right circular cylinder is \( a \) square units, the radius of its base is \( r \) units, and the volume is \( v \) cubic units, then determine the value of \( \cfrac{ar}{v} \).

18. If the curved surface area of a right circular cylinder is \(C\) square units, the radius of the base is \(r\) units, and the volume is \(V\) cubic units, then determine the value of \(\cfrac{Cr}{V}\).

19. If the total surface area of a cube is \(s\) square units and the length of its diagonal is \(d\) units, what is the relationship between \(s\) and \(d\)?

(a) s=6d\(^2\) (b) 3s=7d (c) s\(^3\)=d2\(^2\) (d) d\(^2\)=s/2

20. If the curved surface area of a right circular cylinder is \(c\) square units, the radius of the base is \(r\) units, and the volume is \(v\) cubic units, then find the value of \(\cfrac{cr}{v}\).

21. 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.

22. Let the length of each edge of the cube be \(a\) cm \(\therefore\) The total surface area of the cube = \(6a^2\) square cm If the edge length is increased by 20%, the new edge length = \(a + a \times \cfrac{20}{100}\) cm \(= a + \cfrac{a}{5} = \cfrac{6a}{5}\) cm \(\therefore\) New total surface area of the cube = \(6\left(\cfrac{6a}{5}\right)^2\) square cm = \(\cfrac{216a^2}{25}\) square cm \(\therefore\) Percentage increase in surface area \(= \cfrac{\cfrac{216a^2}{25} - 6a^2}{6a^2} \times 100\%\) \(= \cfrac{(216a^2 - 150a^2) \times 100}{25 \times 6a^2}\%\) \(= \cfrac{4 \times 66a^2}{6a^2}\%\) \(= 44\%\)

23. If the volume of a cube is \(V\) cubic centimeters, the total surface area is \(S\) square centimeters, and the length of the diagonal is \(d\) centimeters, then prove that \(Sd = 6\sqrt{3}V\).

24. The lateral surface area of a right circular cone is \( \sqrt{5} \) times the area of its base. What is the ratio of the cone’s height to the radius of its base?

25. If a solid sphere is melted to form a solid right circular cylinder, then the _____ of the sphere and the cylinder will be equal.

26. The lateral surface area of a right circular cone is \(\sqrt{5}\) times the area of its base. What is the ratio of its height to diameter?

27. If a right circular cone has a radius of \(\frac{r}{2}\) units and a slant height of \(2l\) units, then the total surface area is \(2πr(r + l)\).

28. A solid cuboid has a length, width, and height ratio of \(4:3:2\), and its total surface area is \(468\) square cm. Determine the volume of the cuboid.

29. If the radius of a hemisphere is \(2r\) units, then the total surface area will be _____.

30. If a right circular cone has a volume of \(x\) cubic units and a base area of \(y\) square units, then its height is _____.