1. 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?

2. The curved surface area of a right circular cone is \(\sqrt{10}\) times the area of its base. What is the ratio of the height of the cone to the diameter of its base?

3. 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 _____.

4. If the curved surface area of a right circular cone is \(\sqrt{17}\) times its base area, what is the ratio of the cone’s height to the diameter of its base?

5. The curved surface area of a cone is \(\sqrt{5}\) times the area of its base. What will be the ratio of its radius to height?

(a) 1:2 (b) 1:3 (c) 1:4 (d) 1:8

6. The curved surface area of a right circular cone is √5 times the area of its base. What is the ratio of the cone’s height to its radius?

7. "The lateral surface area of a right circular cone is √5 times the area of its base. Write the ratio of the height to the radius of the cone."

8. The curved surface area of a right circular cone is \(\sqrt{10}\) times the area of its base. Show that the height of the cone is three times the radius of its base.

9. The lateral surface area of a cone is √5 times the area of its base. Determine the ratio of the cone’s height to its radius.

10. If the lateral surface area of a right circular cone is √5 times the area of its base, what is the ratio of its height to its radius?

11. The lateral surface area of a right circular cone is √5 times its base area. Determine the ratio of the cone's height to its radius.

12. 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.

13. If the radius of a right circular cone is \(\cfrac{r}{2}\) units and the slant height is \(2l\) units, then the total surface area is –

(a) \(2πr (l+r)\) square units (b) \(πr\left(l+\cfrac{r}{4}\right)\) square units (c) \(πr(l+r)\) square units (d) \(2πr l\) square units

14. 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 \(2πr(r + l)\).

15. If the radius of a right circular cone is \(\cfrac{r}{2}\) units and the slant height is \(2l\) units, then the total surface area is –

(a) \(2πr (l+r)\) square units (b) \(πr\left(l+\cfrac{r}{4}\right)\) square units (c) \(πr(l+r)\) square units (d) \(2πr l\) square units

16. 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.

17. The slant height of a right circular cone is 7 cm, and the total surface area is 147.84 cm². Determine the radius of its base.

18. The slant height of a right circular cone is 7 cm, and its total surface area is 147.84 square cm. Find the radius of its base.

19. If the radius of a tall circular cone is \( \frac{r}{2} \) units and the slant height is \( 2l \) units, then find the total surface area of the cone.

(a) \(2πr(l+r)\) square units (b) \(πr(l+\cfrac{r}{4})\) square units (c) \(πr(l+r)\) square units (d) \(2πrl\) square units

20. 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 \[ \left( \frac{1}{h^2} + \frac{1}{r^2} \right) \]

21. 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.

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

23. Find the radius of the base of a right circular cone if its slant height is 7 cm and its total surface area is 147.84 cm².

24. A rectangle has a length of \( a \) units and a breadth of \( b \) units. When the rectangle is wrapped to form a right circular cylinder, the circumference of the base of the cylinder is equal to the length of the rectangle. The lateral surface area of the cylinder will be _____.

25. 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 \(\cfrac{1}{h^2}+\cfrac{1}{r^2}\).

26. The slant height of a right circular cone is 7 cm and its total surface area is 147.84 square cm. Find the radius of the base and the area of the base of the cone. Let me know if you'd like the full solution worked out as well. I'm happy to help.

27. The slant height of a right circular cone is 7 cm and its total surface area is 147.84 sq. cm. Find the radius of the base of the cone.

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

29. A rectangular piece of soft cardboard has a length that is four times its width. When rolled along its length, it forms a right circular cone whose lateral surface area is 100 square centimeters. What is the height of the cone?

30. 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.