1. 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.

2. 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.

3. 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?

4. To build a memorial platform measuring 350 cm in length, 200 cm in width, and 150 cm in height, how many bricks of dimensions 25 cm in length, 12.5 cm in width, and 7.5 cm in height will be required?

(a) 4000 (b) 4280 (c) 4480 (d) 4844

5. A rectangular container has a base in the shape of a rectangle with a length of 60 cm and a width of 45 cm. The container has a height of 20 cm and is half-filled with water. Determine the side length of a metallic cube that, when placed in the container, will cause the water level to reach the brim.

6. "Four cylindrical concrete pillars each have a diameter of 17.5 cm and are to be plastered all around with a 3.5 cm thick layer of sand-cement mixture. If the height of each pillar is 3 metres, calculate and write how many cubic decimetres of mortar will be required."

7. A hollow vertical cylindrical iron pipe has an outer radius of 5 cm and an inner radius of 4 cm. If the total surface area of the pipe is 1188 square cm, what is the length of the pipe?

8. Several coins, each with a diameter of 1.5 cm and thickness of 0.2 cm, are melted to form a solid vertical cylindrical rod with a height of 10 cm and diameter of 4.5 cm. Determine the number of coins used.

9. To raise a plot of land measuring 25.5 meters in length and 20.2 meters in width by filling it with soil up to a height of 1.5 meters, a pit is dug in the adjacent plot with dimensions 15.3 meters in length and 10.1 meters in width. What will be the depth of the pit?

(a) 5 meters (b) 10 meters (c) 15 meters (d) 20 meters

10. A hollow sphere made of lead sheets with a thickness of 1 cm has an outer radius of 6 cm. If the sphere is melted to form a solid cylindrical rod with a radius of 2 cm, what will be the length of the rod?

11. A hollow sphere is made of lead sheet 1 cm thick and has an outer radius of 6 cm. If the sphere is melted and recast into a right circular cylinder with a radius of 2 cm, what will be the length of the cylinder? Let me know if you'd like the full solution too. I'm ready when you are.

12. "A team of firefighters used a water-filled cylindrical tanker to extinguish a fire by spraying water through three hosepipes, each with a diameter of 2 cm, at a rate of 420 metres per minute for 40 minutes. If the diameter of the tanker is 2.8 metres and its length is 6 metres, then (i) calculate how much water was used to extinguish the fire, and (ii) determine how much water remains in the tanker."

13. "Four cylindrical concrete pillars each have a diameter of 17.5 cm and are to be plastered all around with a 3.5 cm thick layer of sand-cement mixture. If the mortar is prepared by mixing sand and cement in a 4:1 ratio, calculate and write how many cubic decimetres of cement will be required."

14. 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.

15. A vertical cylindrical tube made of iron is open at both ends. Its height is 42 cm, thickness is 1 cm, and the outer radius is 10 cm. Calculate the volume of iron used to make the tube.

16. We take a vertical cylindrical drum with radius 21 cm and height 21 cm, and a solid iron sphere with a diameter of 21 cm. We calculate the ratio of the volumes of the drum and the solid iron sphere (neglecting the thickness of the drum). Next, we completely fill the drum with water and then fully immerse the sphere into it. Now, determine the new depth of water in the drum after the sphere is removed.

17. 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.

18. 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) \(2x\)

19. In a circle with center O, a tangent PT is drawn from an external point P to the circle, with T being the point of tangency. If PT = 12 cm and OP = 13 cm, the diameter of the circle will be:

(a) 5 cm (b) 8 cm (c) 6 cm (d) 10 cm

20. 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

21. 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

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. A solid cylindrical iron rod has a cross-sectional diameter of 16 cm and a length of 1 meter. If this rod is melted and used to form right circular cones, each with a height of 4 cm and a base radius of 5 cm, how many such cones can be made?

24. A hollow cube is made using copper sheets that are 1.5 cm thick, with an external edge length of 20 cm. After melting the cube, how many solid cuboids measuring 7 cm × 3 cm × 1 cm can be made from it?

(a) 200 (b) 220 (c) 100 (d) 147

25. A hollow cylindrical iron pipe open at both ends has an inner radius and outer radius of 4 cm and 5 cm respectively. If the total surface area of the pipe is 1188 square centimeters, what is the length of the pipe?

26. 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?

27. A vertical cylindrical tank with a height of 5 meters is completely filled with water. If water flows out through a pipe with a diameter of 8 cm at a speed of 225 meters per minute, and the entire tank empties in 45 minutes, then find the diameter of the tank.

28. A vertical cylindrical gas jar with a diameter of 7 cm contains some water. If a solid iron right circular cylindrical piece with a diameter of 5.6 cm and height 5 cm is completely submerged in the water, calculate how much the water level will rise.

29. The rectangular village field has a length of 20 meters and a width of 15 meters. To install pillars at the four corners, four cube-shaped holes of 4 meters in length were dug and the removed soil was spread evenly over the remaining area of the field. Calculate how much the surface level of the field will rise.

30. AOB is the diameter of a circle centered at O. C is a point on the circumference. If AC = 3 cm and BC = 4 cm, then the length of AB will be _____.