SPI Mathematics · Mensuration 3D

Melting, Recasting and Water Level

When a solid is melted and remade, or dropped into water, its volume stays the same: equate the old volume to the new one and solve.

All Mensuration 3D notes

Why this matters

Six questions in the bank, five MODERATE and one EASY, one of them from the 2022 paper. Coins from a cuboid or a cylinder, spheres from a cylinder, and the rise of water when a ball is dropped in. Each is one division of two volumes from the earlier pages.

Concept 1 of 2: Melting and recasting: how many pieces

Melting changes the shape, never the amount of metal. Find the volume you start with, find the volume of one new piece, and divide.

Definition

Volume is kept: old volume $=$ total new volume.
Number of pieces $=$ big volume $\div$ volume of one piece.
A coin is a short cylinder: its radius is half its diameter and its height is its thickness.
When both solids are round, write both volumes with $\pi$ and cancel it before dividing.
If your count is not among the options, check the radius and the units once, then choose 'None of these'.

Recasting

V_{\text{old}} = n \times V_{\text{piece}}, \qquad n = \frac{V_{\text{old}}}{V_{\text{piece}}}

Worked example

A metal cylinder of radius 4 cm and height 12 cm is melted and made into spheres of radius 2 cm. How many spheres are made?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q43Moderate

Example 1 · Mensuration 3D · Melting, Recasting and Water Level

Dimensions of a cuboid are 33cm x 18cm x 8cm. If this cuboid is melted to mint coins of diameter 6cm and thickness of 0.2cm then____coins can be obtained.

Using the coin's diameter as its radius

A coin of diameter 3 cm has radius 1.5 cm. Using 3 makes each coin four times too big and the count four times too small.

Same shape, smaller size: cube the ratio

Small spheres from a big sphere:

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

. A sphere of radius 4 cm makes 64 balls of radius 1 cm, not 4 and not 16.

Drill 3 more on melting and recasting: how many pieces

Concept 2 of 2: Rise in water level

Push a stone under water and the water has to go somewhere: it rises up the sides. The water that rises is a thin slab with the vessel's base and a volume equal to the stone's.

Definition

A solid fully under water pushes up water of its own volume.
Rise $=$ volume of the solid $\div$ base area of the vessel.
In a cylindrical vessel of radius $R$ : $\pi R^2 \times \text{rise} = V_{\text{solid}}$ .
How full the vessel was at the start does not matter, as long as the solid is covered and nothing spills.

Water rise in a cylinder

\pi R^2 \times \text{rise} = V_{\text{solid}}

Rradius of the vessel (not of the solid)

Worked example

Ten marbles, each of radius 1.5 cm, are dropped into a cylindrical beaker of radius 5 cm partly filled with water. By how much does the water rise?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Mensuration 3D · Melting, Recasting and Water Level

A sphere of radius 6 cm is dropped into a cylindrical vessel partly filled with water. The radius of the vessel is 8 cm. If the sphere is submerged completely, then the surface of the water rises by

Dividing by the wrong circle

The risen water spreads over the vessel's base, so divide by

\pi R^2

of the vessel, not by the area of the ball or cone dropped in.

Worrying about the starting depth

'Partly filled' only tells you the solid is covered. The rise depends on the solid's volume and the vessel's base, not on how much water was there.

Drill 1 more on rise in water level

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 (2)

Melting and recasting: how many pieces
Recasting
$V_{\text{old}} = n \times V_{\text{piece}}, \qquad n = \frac{V_{\text{old}}}{V_{\text{piece}}}$
Rise in water level
Water rise in a cylinder
$\pi R^2 \times \text{rise} = V_{\text{solid}}$

Watch out for (4)

Using the coin's diameter as its radius
→ Melting and recasting: how many pieces
Same shape, smaller size: cube the ratio
→ Melting and recasting: how many pieces
Dividing by the wrong circle
→ Rise in water level
Worrying about the starting depth
→ Rise in water level

Test yourself on a real paper

Sit a past SPI paper, timed and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Sit an SPI past paper Drill all 6 questions