1. The shape of a one-end sharpened pencil is a combination of a cone and a **cylinder**.
2. The shape of a one-end sharpened pencil is a combination of a cone and a ——.
3. The shape of a one-ended sharpened pencil is a combination of a cone and a __________.
4. The shape of a sharpened pencil is a combination of a cone and a _____.
5. The shape of a pencil is a combination of a cone and a ___.
6. The shape of a single-cut pencil is a combination of a cone and _____.
7. A hollow iron cylinder of height 20 cm has an outer radius of 5 cm and an inner radius of 4 cm. This cylinder is melted and recast into a solid cone whose height is one-third of the original cylinder's height. Find the diameter of the base of the cone.
8. A one-end open cylindrical pipe is 12 meters long and has a radius of 4.5 meters. What is the length of the longest rod that can be placed inside the cylinder?
9. The mountain is in the shape of a right circular cone. Its slant height is 2.5 km, and its base area is 1.54 square km. Determine the height of the mountain.
10. A right circular cylinder with the same radius and height has a volume that is three times the volume of a right circular cone.
11. 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.
12. A solid right circular cone is melted and recast into a solid right circular cylinder with the same radius and a height of 5 cm. Find the height of the cone.
(a) 10 cm (b) 15 cm (c) 18 cm (d) 24cm
13. 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.
14. The bottom part of a solid is in the shape of a hemisphere, and the top part is in the shape of a right circular cone. If the surface areas of both parts are equal, then calculate and write the ratio of the radius to the height of the cone.
15. If the ratio of the radii of two perpendicular solid cones is 2:3 and the ratio of their heights is 5:3, what is the ratio of their volumes ?
(a) 27:20 (b) 20:27 (c) 4:9 (d) 9:4
16. What is the ratio of the volumes of a right circular cylinder and a cone having the same radius and height?
(a) 1:3 (b) 3:2 (c) 3:1 (d) 2:3
17. A solid vertical cylindrical cone has both its diameter and height equal to 21 cm. Find the volume of the largest possible sphere that can be obtained from this cone. Also, determine the ratio of the volumes of the cone and the sphere.
18. What is the ratio of the volumes of a solid right circular cylinder, a solid right circular cone, and a solid sphere, all having the same diameter and the same height?
19. 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."
20. A right circular cylinder made of iron sheet is open at both ends. Its height is 42 cm, thickness is 1 cm, and its outer radius is 10 cm. Find the amount of iron used to make the cylinder.
21. In a certain business, A's capital is one and a half times that of B's capital. If B receives ₹1,500 as profit at the end of the year, how much profit will A receive?
22. A and B started a joint business by investing ₹6,200 and ₹10,000 respectively. They agreed that A would receive 20% of the profit for managing the business, and 10% of the remaining profit would be kept aside as savings. The rest of the profit would be divided between A and B in proportion to their capital investments. If the total profit at the end of the year is ₹45,000, how much money will A receive in total?
23. A right circular cylinder and a cone have equal bases, and the ratio of their volumes is 3:2. Prove that the height of the cone is half the height of the cylinder.
24. There is some water in a vertical cylindrical drum. Three solid right circular cone-shaped iron pieces, each having a base diameter of 32 cm and height of 40 cm, are completely submerged in the water. As a result, the water level in the drum rises by 12.8 cm. Find the diameter of the drum.
25. If the base diameter of a cone is 21 cm and its slant height is 17.5 cm, find the volume of the cone.
26. If one side of a rhombus is 6 cm and one of its angles measures 60°, then the area of the rhombus-shaped field is –
(a) \(9\sqrt{3}\) square cm (b) \(18\sqrt{3}\) square cm (c) \(36\sqrt{3}\) square cm (d) \(6\sqrt{3}\)square cm
27. If the length of one diagonal of a rhombus is three times the length of the other diagonal, and the area of the rhombus-shaped field is 96 square centimeters, then the length of the longer diagonal is –
(a) 8 cm (b) 12 cm (c) 16 cm (d) 24cm
28. The area of a trapezium-shaped field is 162 square centimeters and its height is 6 cm. If one of the parallel sides of the trapezium is 23 cm long, then the length of the other parallel side is –
(a) 29 cm (b) 31 cm (c) 32 cm (d) 33 cm
29. Shuvendu and Noushad started a business by investing ₹1500 and ₹1000 respectively. After one year, the business incurred a loss of ₹75. Shuvendu's share of the loss is –
(a) ₹ 45 (b) ₹ 30 (c) ₹ 25 (d) ₹ 40
30. If the volume of a right circular cone is \(100\pi\) cubic cm and its height is 12 cm, find the slant height of the cone.
