Volume Applications • Topic 1 of 3

Melting & Recasting

When a solid is melted and reshaped, only the shape changes — the volume is conserved. So the whole method is one equation: volume of the original solid = volume of the new solid (or n times the volume of each small piece). The mass and material stay the same, so set V(old) = V(new) and solve for the unknown dimension. The CAT-smart move is to cancel before you multiply: π appears on both sides and disappears, fractions like 4/3 and 1/3 cancel against each other, and you are left with a clean ratio of cubes or a simple linear equation. A sphere of radius R melted into n small spheres of radius r gives R³ = n·r³, so the radius ratio is the cube root of n — never the number itself. When a sphere is drawn into a long thin wire (a cylinder), equate (4/3)πR³ = πr²h and the length h pops out. Always check units agree before equating.

✅ Solved examples

1. A solid metal sphere of radius 6 cm is melted and recast into a cylinder of radius 4 cm. Find the height of the cylinder.
Volume conserved: (4/3)π(6)³ = π(4)²h ⇒ (4/3)(216) = 16h ⇒ 288 = 16h ⇒ h = 18 cm.
2. A metallic sphere of radius 10.5 cm is melted into small cones each of radius 3.5 cm and height 3 cm. How many cones are formed?
Sphere V = (4/3)π(10.5)³. Cone V = (1/3)π(3.5)²(3) = (1/3)π(12.25)(3) = 12.25π. Sphere V = (4/3)π(1157.625) = 1543.5π. Count = 1543.5π / 12.25π = 126 cones.
3. A cube of edge 12 cm is melted and recast into 8 equal cubes. Find the edge of each small cube.
V(big) = 12³ = 1728. Each small = 1728/8 = 216 ⇒ edge = ∛216 = 6 cm.
4. A solid sphere of radius 3 cm is drawn into a wire of radius 0.2 cm. Find the length of the wire (in cm).
(4/3)π(3)³ = π(0.2)²·h ⇒ (4/3)(27) = 0.04h ⇒ 36 = 0.04h ⇒ h = 900 cm.

✏️ Practice — try these, take hints as needed

1. A sphere of radius 6 cm is melted into n spheres of radius 2 cm. Find n.
Volumes conserve; π and 4/3 cancel.
n = R³ / r³.
6³ / 2³ = 216/8.
27
2. A cylinder of radius 6 cm and height 16 cm is melted into a sphere. Find the sphere’s radius.
πr²h = (4/3)πR³.
36×16 = (4/3)R³.
R³ = 432 ⇒ R = ∛432.
6∛2 cm (≈ 7.56 cm)
3. A cone of radius 6 cm and height 24 cm is melted into a sphere. Find the sphere’s radius.
(1/3)πr²h = (4/3)πR³.
(1/3)(36)(24) = (4/3)R³.
288 = (4/3)R³ ⇒ R³ = 216.
6 cm
4. A metal cuboid 9 cm × 8 cm × 2 cm is recast into a cube. Find the cube’s edge.
Volume = 9×8×2.
Edge = ∛144... recheck product.
144 is not a perfect cube; V = 144 ⇒ ∛144.
∛144 cm ≈ 5.24 cm
5. A sphere of radius 9 cm is melted and recast into a cone of height 27 cm. Find the cone’s base radius.
(4/3)π(9)³ = (1/3)πr²(27).
(4/3)(729) = 9r².
972 = 9r² ⇒ r² = 108.
6√3 cm (≈ 10.39 cm)

📝 Topic test — 8 questions

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