Melting and Recasting of Solids Using Volume Conservation
Melting and Recasting of Solids deals with changing one solid into another without losing material. The total volume remains constant when a solid is melted and recast. Questions usually involve cubes, cuboids, cylinders, spheres, cones, hollow solids, volume ratios and the number of smaller articles formed.
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What is Melting and Recasting of Solids?
The basic equation is Original volume = Total recast volume. If one solid forms n identical smaller solids, then V_original = n × V_each. For example, a cube of side 6 cm is recast into cubes of side 2 cm: n = 6³ ÷ 2³ = 216 ÷ 8 = 27 cubes.
Melting and Recasting of Solids Video Lessons
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Melting and Recasting: Volume Conservation
Understand how volume remains conserved when a solid is melted and recast into another shape, and apply this principle to solve related mensuration problems.
Melting and Recasting of Solids Formula & Tricks
Important Formulas
Melting changes the shape but not the amount of material.
Use this when one solid is converted into several identical solids.
Select the volume formula according to the original and recast shapes.
For a hollow cylinder, V_material = πh(R² − r²), where R and r are the outer and inner radii.
This rule gives the number of smaller similar solids formed from one larger similar solid.
Quick Tricks
For similar solids, the number of smaller solids is the cube of the ratio of corresponding lengths.
When both volume expressions contain π, cancel it before solving. This often avoids decimal calculations.
Convert metres, centimetres and millimetres before finding volume. A volume conversion uses the cube of the length conversion.
Melting and Recasting of Solids Concepts
Conservation of Volume
If a solid is changed into n pieces, write V_original = V₁ + V₂ + ... + Vₙ. For identical pieces, this becomes V_original = nV_piece. Surface area is not generally conserved because the shape changes.
Recasting One Solid into Another
For a cylinder recast into a cone, πr₁²h₁ = πr₂²h₂/3. For a sphere recast into a cylinder, 4πR³/3 = πr²h. Use the same unit for every length before equating the volumes.
Number of Similar Smaller Solids
If the corresponding side ratio is L:l, then n = (L/l)³. This applies to cubes, spheres and other similar three-dimensional solids. The ratio must compare corresponding lengths, not areas.
Hollow Solids and Material Volume
For a hollow cylinder with outer radius R, inner radius r and height h, material volume = πh(R² − r²). This volume is then equated to the volume of the solid formed after recasting.
Change in Dimensions After Recasting
For a cuboid formed from a solid of volume V, the unknown length is V divided by the product of the known breadth and height. For a cube, its side is the cube root of V.
Melting and Recasting of Solids Visual Guide & Data
Use the table and chart below to understand the data better.
Volume formulas used in recasting problems
Common volume expressions for solids appearing in melting and recasting questions.
| Solid | Volume |
|---|---|
| Cube | a³ |
| Cuboid | lbh |
| Cylinder | πr²h |
| Cone | πr²h/3 |
| Sphere | 4πr³/3 |
Note: Use consistent units before substituting values.
Equal volume before and after recasting
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Melting and Recasting of Solids: Revision Points
Use these rules to solve volume-conservation questions quickly.
- Melting and recasting preserve volume, not surface area.
- Write original volume = total recast volume.
- For n identical pieces, n = original volume ÷ volume of one piece.
- For similar solids, number of pieces = cube of the corresponding length ratio.
- For hollow solids, subtract inner volume from outer volume.
- Cancel common factors such as π before solving.
- Convert every measurement to the same unit before calculating volume.
- A volume conversion is cubic: 1 m³ = 1,000,000 cm³.
Melting and Recasting of Solids FAQs
What is the main formula for melting and recasting of solids?
The main formula is V_original = V_total after recasting. For n identical pieces, V_original = n × V_each.
Are surface areas equal before and after recasting?
No. Volume remains constant because the material is conserved, but surface area usually changes when the shape changes.
A cube of side 9 cm is recast into cubes of side 3 cm. How many cubes are formed?
Number = (9/3)³ = 3³ = 27 cubes.
How is a hollow cylinder handled in a recasting problem?
Use the material volume, not the outer volume alone: V = πh(R² − r²), where R is the outer radius and r is the inner radius.
A cylinder of radius 4 cm and height 6 cm is recast into a cone of radius 4 cm. What is the cone's height?
π×4²×6 = π×4²×h/3, so h = 18 cm.
What happens to volume when every dimension is doubled?
The volume becomes 2³ = 8 times the original volume. Therefore, one larger similar solid can contain the volume of eight smaller solids whose corresponding dimensions are half as large.