1. The length, width, and height of a cuboidal room are respectively \(a\), \(b\), and \(c\) units, and given that \(a + b + c = 25\) and \(ab + bc + ca = 240.5\), what is the length of the longest rod that can be placed inside the room?

2. If the length, breadth, and height of a rectangular room are 5 m, 4 m, and 3 m respectively, calculate and write the length of the longest rod that can be placed inside the room.

3. In a cuboid-shaped room, the length, breadth, and height are respectively \(a\), \(b\), and \(c\) units. Given \(a + b + c = 25\) and \(ab + bc + ca = 240.5\), calculate the length of the longest rod that can be placed inside the room.

4. If a right circular cylinder has a radius of 3 cm and a height of 4 cm, then what is the length of the longest rod that can be placed inside it?

5. If the internal length, width, and height of a rectangular box are 6 meters, 3 meters, and 2 meters respectively, what is the length of the longest rod that can be placed inside the box?

6. If the diameter of a right circular cylinder is 3 cm and its height is 4 cm, then the longest rod that can be placed inside the cylinder will have a length of ___________ cm.

7. The length, width, and height of a rectangular block are 11 cm, 9 cm, and 6 cm respectively. How many coins of 3 cm diameter and ¼ cm thickness can be made from this block?

8. In a rectangular prism, the ratio of length, width, and height is \(4:5:3\). If the length of one diagonal is \(35\sqrt2\) cm, what is the total surface area?

9. The radius of a solid right circular iron rod is 32 cm and its length is 35 cm. If the rod is melted and recast into solid cones each with a radius of 4 cm and height of 28 cm, calculate how many such cones can be made.

10. The sum of the length, width, and height of a rectangular box is 24 cm, and the length of its diagonal is 15 cm. What is the total surface area of the rectangular box?

(a) 360 square cm (b) 221 square cm (c) 351quare cm (d) 256 quare cm

11. 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?

12. A vertical hollow cylindrical pipe has an outer radius of 16 cm and an inner radius of 12 cm, with a height of 36 cm. If the pipe is melted and solid cylindrical rods are made from it, each with a diameter of 2 cm and a length of 6 cm, determine how many such solid rods can be produced.

13. The sum of the length, width, and height of a right rectangular prism is 25 cm, and its total surface area is 264 square cm. Determine the length of its diagonal.

14. Translate the following: "The internal length, width, and height of a tea box are 7.5 decimeters, 6 decimeters, and 5.4 decimeters respectively. The weight of the box when filled with tea is 52 kilograms 350 grams. But when empty, the box weighs 3.75 kilograms. Determine the weight of tea per cubic decimeter."

15. ABC and POR are two similar triangles. If BC = 5 cm, QR = 4 cm, and the height AD = 3 cm, then what is the length of the height PE?

(a) 4.2 cm (b) 1.25 cm (c) 5.4 cm (d) 2.4 cm

16. The sum of the length, breadth, and height of a rectangular box is 24 cm, and the length of its diagonal is 15 cm. What is the total surface area of the box?

17. A classroom needs to be built for 40 students in such a way that each student gets 5 square meters of floor area and 25 cubic meters of space. If the length of the room is 20 meters, what should be its height?

(a) 5 meters (b) 7 meters (c) 9 meters (d) 11 meters

18. A solid cylindrical iron rod has a base radius of 32 cm and a height of 35 cm. If the rod is melted and recast into solid cones, each with a radius of 8 cm and a height of 28 cm, how many such cones can be made?

19. Gautam and Samaresh started a joint business with ₹12,500 and ₹8,500 respectively. They agreed that 40% of the profit would be equally distributed between them, and the remaining profit would be shared in proportion to their investments. If Gautam received ₹1,950 as his share of the profit, determine Samaresh’s profit amount.

20. The internal length, breadth, and height of a tea box are 7.5 decimeters, 6 decimeters, and 5.4 decimeters respectively. The weight of the tea-filled box is 52 kg 350 g. If the weight of the empty box is 3.75 kg, calculate and write the weight of tea per cubic decimeter.

21. If the length, breadth, and height of a packet containing one gross of matchboxes are 2.8 dm, 1.5 dm, and 0.9 dm respectively, calculate the volume of one matchbox. [One gross = 12 dozen = 144 matchboxes] But if the length and breadth of a single matchbox are 5 cm and 3.5 cm respectively, calculate and write its height.

22. A rectangular pit has a length of 40 meters, a width of 12 meters, and a depth of 16 meters. Can a wooden plank measuring 5 meters in length, 4 meters in width, and 2 meters in thickness be placed inside the pit?

(a) 190 (b) 192 (c) 184 (d) 180

23. Three solid cubes have edge lengths of 3 cm, 4 cm, and 5 cm respectively. These cubes are melted and recast into a new solid cube. Write the length of an edge of the new cube.

24. The lengths of two adjacent walls of a room are 12 m and 8 m respectively. If the height of the room is 4 m, calculate the floor area of the room.

25. The dimensions of a cuboid are 12 cm, 6 cm, and 3 cm respectively. Calculate the length of each edge of a cube whose volume is equal to that of the cuboid.

26. Two friends started a joint business with investments of ₹40,000 and ₹50,000 respectively. They agreed that 50% of the profit would be shared equally between them, and the remaining 50% would be divided in proportion to their capital contributions. If the first friend’s profit share is ₹800 less than the second friend’s, calculate the amount received by the first friend.

27. Kaushik and Sanjay started a joint business by investing ₹12,500 and ₹8,500 respectively. They agreed that 40% of the profit would be equally distributed between them, and the remaining profit would be shared in the ratio of their capitals. If Kaushik received ₹1,950 as profit, what was Sanjay's share of the profit?

28. 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) \(18\pi \, cm^2\)

29. If three metallic cubes with edges measuring 3 cm, 4 cm, and 5 cm are melted and recast into a new cube, the length of the edge of the new cube will be –

(a) 6 cm (b) 7 cm (c) 8 cm (d) 9 cm

30. The volume of a right rectangular prism is 960 cubic centimeters. If the ratio of its length, width, and height is 6:5:4, what is the total surface area of the prism?

(a) 590 square cm (b) 592 square cm (c) 295 square cm (d) 596 square cm