1. If the radius and height of a cone are each doubled, what will be the volume of the new cone in relation to the volume of the original cone? (a) 3 times (b) 4 times (c) 6 times (d) 8 times

2. If a right circular cone has a base diameter of 7 cm and a slant height of 10 cm, what is the total surface area of the cone? (a) 184.5 square cm (b) 185.4 square cm (c) 148.5 square cm (d) 184.6 square cm

3. Let radius \(r\), slant height \(l = 7\) m, and total canvas area \(= 77\) m² Lateral surface area \(= \pi r l = 7\pi r\) Base area \(= \pi r^2\) Total area \(= 7\pi r + \pi r^2 = 77\) \(\pi r(r + 7) = 77\) \(r(r + 7) = \frac{77}{\pi} \approx 24.52\) \(r^2 + 7r - 24.52 = 0\) \(r = \frac{-7 \pm \sqrt{147.08}}{2} \approx \frac{-7 \pm 12.13}{2}\) \(r \approx 2.565\) Base area \(= \pi r^2 \approx 3.1416 \times 6.58 \approx 20.67\) m² (a) 38.5 square meters (b) 39.5 square meters (c) 36.5 square meters (d) 37.5 square meters

4. 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) \(18\pi \, cm^2\)

5. A right circular cone has a base radius of 20 cm and a height of 10 cm. If it is melted down, how many solid spheres of radius 2 cm can be made from it? (a) 200 (b) 100 (c) 250 (d) 125

6. The volume of a right circular cone is V. If both the height and the radius are doubled, the new volume will be: (a) 3V (b) 4V (c) 6V (d) 8V

7. If a cone has a base diameter of 12 cm and an apex angle of 60°, what is its height? (a) \(2\sqrt3\) cm (b) \(6\) cm (c) \(6\sqrt3\) cm (d) None of the above

8. If the base area of a right circular cone is \(A\), its height is \(H\), and its volume is \(V\), then express \(H\) in terms of \(V\) and \(A\). (a) \(H=\cfrac{3V}{A}\) (b) \(H=\cfrac{V}{A}\) (c) \(H=\cfrac{V}{3A}\) (d) \(H=\cfrac{3V}{2A}\)

9. If the diameters of two cones are equal and the ratio of their slant heights is 5:7, then what is the ratio of their curved surface areas? (a) 25:7 (b) 25:49 (c) 5:49 (d) 5:7

10. If the volume of a right circular cone is \(100\pi\) cubic centimeters and its height is 12 cm, then what is its slant height? (a) 13 cm (b) 16 cm (c) 9 cm (d) 26 cm

11. If the volumes of two cones are in the ratio 2:3 and the ratio of their base radii is 1:2, then what is the ratio of their heights? (a) 3:8 (b) 8:3 (c) 3:4 (d) 4:3

12. 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

13. If both the height and radius of a cone are doubled, the volume of the cone will increase – (a) tripled (b) five times (c) seven times (d) nine times

14. If a right circular cone has a height of 21 cm and a radius of 12 cm, its volume will be – (a) 4710 cubic cm (b) 9504 cubic cm (c) 3168 cubic cm (d) None of the above

15. If a right circular cone has a height of \(12\) cm and a volume of \(1024\pi\) cubic cm, what is the slant height of the cone? (a) 20 cm (b) 21 cm (c) 22 cm (d) 23 cm

16. If the radius of a right circular cone is kept constant and its height is increased by 10%, by what percentage will the volume increase? (a) 10% (b) 11% (c) 12% (d) 14%

17. 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

18. 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

19. If the volumes of two cones are in the ratio 1:4 and the ratio of the diameters of their bases is 4:5, find the ratio of their heights (a) 1:5 (b) 5:4 (c) 5:16 (d) 25:64

20. If the radius and height of a cone are both doubled, the volume of the cone will increase to— (a) 3 times (b) 4 times (c) 6 times (d) 8 times

21. The ratio of the base radii of two right circular cones is 2 : 3, and the ratio of their heights is 3 : 2. Then the ratio of their volumes is — (a) 3:2 (b) 9:4 (c) 4:9 (d) 2:3

22. 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

23. If the volume ratio of two right circular cones is 1:4 and the ratio of their base radii is 4:5, then the ratio of their heights is—? (a) 1:5 (b) 5:4 (c) 25:16 (d) 25:64

24. 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. True / False

25. The height, radius, and slant height of a right circular cone always form the three sides of a right-angled triangle. True / False

26. The height of a right circular cone is twice the length of its radius. If the height were 7 times the base diameter, the volume of the cone would be 539 cubic cm more. Find the height of the cone.

27. 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?

28. If the volume of a right circular cone is V, the base radius is R, and the height is H, then H = _____

29. 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.

30. If the volume of a right circular cone is \(X\) cubic units, the area of its base is \(Y\) square units, and its height is \(Z\) units, then what is the value of \(\frac{YZ}{X}\)?

31. If the volume of a cone is \(x\), the base area is \(y\), and the height is \(z\), then the value of \(\cfrac{x}{yz}\) will be 3.

32. 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.

33. If a right circular cone has a curved surface area of \(154\sqrt{2}\) square cm and a base radius of 7 cm, determine its apex angle.

34. 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.

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

36. 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.

37. A vertical conical tent can accommodate 11 people. Each person requires 4 square meters of ground space and 20 cubic meters of air. Determine the height of the tent designed specifically for those 11 people.

38. 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?

39. In a tall circular conical tent, 11 people can stay. Each person requires 4 square meters of floor space and 20 cubic meters of air. Determine the height of the tent designed specifically for these 11 people.

40. A right circular cone has equal height and diameter. Determine the ratio of the curved surface area to the base area of the cone.

41. To make a vertical circular conical tent, 77 square meters of canvas is used. If the slant height of the tent is 7 meters, then what is the area of the base of the tent?

42. 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?

43. 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 \[ h = \frac{\sqrt{S^2 - \pi^2 r^4}}{\pi r} \]

44. If a right circular cone has volume \(V\) cubic units, base area \(A\) square units, and height \(H\) units, then determine the value of \(\cfrac{AH}{3V}\).

45. To make a vertical circular conical tent, \(188\frac{4}{7}\) square meters of cloth is required. If the base perimeter of the tent is \(37\frac{5}{7}\) meters, then what are the slant height, radius, and height of the tent?

46. Determine the radius of the base of a solid right circular cone with a slant height of 7 cm and a total surface area of 147.84 square cm.

47. A right circular cone has a height of 12 cm and a volume of \(100\pi\) cubic cm. Find the slant height of the cone.

48. A hemispherical bowl made of silver foil has an outer diameter of 4 cm and an inner diameter of 4 cm. The bowl is melted down to form a solid cone with a diameter of 4 cm. Find the height of the cone."

49. A right circular cone has a base diameter of 21 meters and a height of 14 meters. If the cost of painting is ₹1.50 per square meter, how much will it cost to paint the curved surface area?

50. The volume of a right circular cone is \(V\) cubic units. If the area of the base is \(A\) square units and the height is \(H\) units, then find the value of \(\frac{AH}{V}\).

51. একটি সমকোণী ত্রিভুজের সমকোণ সংলগ্ন বাহুদুটির দৈর্ঘ্য 15 সেমি ও 20 সেমি। অতিভুজকে স্থির রেখে অতিভুজের চারিদিকে ত্রিভুজটিকে একবার ঘোরালে যে দ্বিত-শঙ্কু উৎপন্ন হবে তার মোট ঘনফল নির্ণয় করো। - translate in english