CDS Mathematics · Mensuration 3D

Melting & Recasting

A solid melted and recast into other shapes keeps its volume: find how many pieces, the new dimension, or how the surface area changes.

All Mensuration 3D notes CDS Mathematics strategy

Why this matters

One of the paper's favourite families, and almost always MODERATE. The whole method is one equation — old volume equals new volume — and the only traps are units (millimetres against centimetres) and forgetting that surface area is NOT conserved.

Concept 1 of 4: Volume is conserved

Melting changes the shape, never the amount of metal. Write the old volume, write the new volume with the unknown in it, and set them equal. Common factors like

\pi

\dfrac43\pi

usually cancel.

Definition

Old volume $=$ total new volume.
Several spheres into one: $R^3 = r_1^3 + r_2^3 + \cdots$ (the $\dfrac43\pi$ cancels).
A hollow solid: use only the metal, e.g. $\dfrac43\pi(R^3 - r^3)$ for a shell.
A thin plate is a very short cylinder: area $\times$ thickness.

Recasting

V_{\text{old}} = V_{\text{new}}

Worked example

Three metal spheres of radii

1

6

and

8

cm are melted into one sphere. Find its radius.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (I) 2019 — Elementary Mathematics · Q66Easy

Example 1 · Mensuration 3D · Melting and Recasting

Three copper spheres of radii 3 cm, 4 cm and 5 cm are melted to form a large sphere. What is its radius ?

Drill 8 more on volume is conserved

Concept 2 of 4: How many pieces?

The number of small pieces is the big volume divided by one small volume. When both are the same shape, the formula's constant cancels and the count is just the ratio of the cubes of the lengths.

Definition

Count $= \dfrac{\text{volume of the source}}{\text{volume of one piece}}$ .
Same shape: count $= \left(\dfrac{R}{r}\right)^3$ .
Convert every length to ONE unit first — mm and cm in the same stem are the usual trap.
Filling containers (a bowl into bottles, a cylinder of ice-cream into cones) is the same count.

Same-shape pieces

n = \left(\frac{R}{r}\right)^3

Worked example

How many lead shots of diameter

3

mm can be made from a sphere of radius

6

cm?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (II) 2021 — Elementary Mathematics · Q72Moderate

Example 2 · Mensuration 3D · Melting and Recasting

How many silver coins, 3.5 cm in diameter and of thickness 4 mm, must be melted to form a cuboid of dimensions 21 cm

\times

11 cm

\times

7 cm ? (take

\pi = \frac{22}{7}

)

Drill 3 more on how many pieces?

Concept 3 of 4: Drawn into a wire

A wire is a long thin cylinder. Its volume is its cross-section times its length, so given one of radius or length, the other follows. Keep the length and the radius in the same unit.

Definition

$\pi\rho^2 L =$ the volume of the source.
Sphere of radius $R$ into a wire of radius $\rho$ : $L = \dfrac{4R^3}{3\rho^2}$ .

Sphere into wire

\pi\rho^2 L = \tfrac43\pi R^3

Worked example

A sphere of radius

6

cm is drawn into a wire of radius

2

mm. Find the length of the wire in metres.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (I) 2021 — Elementary Mathematics · Q62Moderate

Example 3 · Mensuration 3D · Melting and Recasting

A solid sphere of diameter 60 mm is melted to stretch into a wire of length 144 cm. What is the diameter of the wire?

Drill 2 more on drawn into a wire

Concept 4 of 4: Surface area is not conserved

Volume stays; surface grows when you break a solid into smaller pieces. One sphere recast into

n

equal spheres has each radius

n^{-1/3}

of the original, and the total surface grows by the factor

n^{1/3}

Definition

$n$ equal spheres from one: $r = \dfrac{R}{\sqrt[3]{n}}$ ; total surface $= \sqrt[3]{n}\times$ the original.
The reverse, many into one, shrinks the total surface.

One sphere into n

\frac{S_{\text{new}}}{S_{\text{old}}} = \sqrt[3]{n}

Worked example

A ball is recast into

27

equal balls. By what factor does the total surface area change?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2023 · CDS (I) 2023 — Elementary Mathematics · Q69Moderate

Example 4 · Mensuration 3D · Melting and Recasting

A solid iron ball is melted and 64 smaller solid balls of equal size are made using the entire volume of iron. What is the ratio of the surface area of the larger ball to the sum of the surface areas of all the smaller balls ?

Drill 1 more on surface area is not conserved

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

Volume is conserved
Recasting
$V_{\text{old}} = V_{\text{new}}$
How many pieces?
Same-shape pieces
$n = \left(\frac{R}{r}\right)^3$
Drawn into a wire
Sphere into wire
$\pi\rho^2 L = \tfrac43\pi R^3$
Surface area is not conserved
One sphere into n
$\frac{S_{\text{new}}}{S_{\text{old}}} = \sqrt[3]{n}$

Test yourself on Mensuration 3D

20 past CDS questions from this chapter, timed at 24 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Mensuration 3D chapter test Drill all 18 questions