Melting and recasting a solid

Melt it and make something new

A goldsmith melts a gold bar and makes rings. A potter squeezes a ball of clay into a cup. The amount of material does not change when the shape does. So the volume stays the same. That one idea solves every problem of this lesson.

The method.

1. Write the volume of the old solid.
2. Write the volume of the new solid, or of one piece.
3. Set them equal. If there are $n$ pieces, the old volume equals $n$ times the volume of one piece.
4. Solve for the unknown.

When a question asks "how many", divide the old volume by the volume of one piece. The answer should be a whole number. If it is not, you can make only the whole part of it.

Keep the units the same. If a wire has a radius in millimetres and a ball has a radius in centimetres, change one of them first.

Worked examples

Example 1. A cylinder into a rod. A solid cylinder of radius $3\ \text{cm}$ and height $10\ \text{cm}$ is melted and drawn into a thin rod of radius $1\ \text{cm}$. How long is the rod?

1. The old volume. $\pi \times (3\ \text{cm})^2 \times 10\ \text{cm} = 90\pi\ \text{cm}^3$.
2. The new volume. The rod is a cylinder of length $L$: $\pi \times (1\ \text{cm})^2 \times L$.
3. Set them equal. $\pi L = 90\pi$ in the units of cm, so $L = 90\ \text{cm}$.

Notice that the radius was divided by $3$ and the length was multiplied by $9$. A radius counts twice in the volume, since it is squared.

Example 2. A cylinder into cones. A solid cylinder of radius $4\ \text{cm}$ and height $18\ \text{cm}$ is melted and made into $12$ identical cones, each with radius $4\ \text{cm}$. Find the height of each cone.

1. The old volume. $\pi \times (4\ \text{cm})^2 \times 18\ \text{cm} = 288\pi\ \text{cm}^3$.
2. One cone. $288\pi\ \text{cm}^3 \div 12 = 24\pi\ \text{cm}^3$.
3. The height. $\dfrac{1}{3}\pi \times (4\ \text{cm})^2 \times h = 24\pi\ \text{cm}^3$, so $\dfrac{16h}{3} = 24$ and $h = 4.5\ \text{cm}$.

The surface area does not stay the same

Volume is conserved. Surface area is not. Here is why. Melt a ball of radius $R$ and make $n$ equal small balls of radius $r$.

• The volumes match: $n \times \dfrac{4}{3}\pi r^3 = \dfrac{4}{3}\pi R^3$. So $r^3 = \dfrac{R^3}{n}$ and $r = \dfrac{R}{\sqrt[3]{n}}$.
• The total area of the small balls is $n \times 4\pi r^2 = 4\pi R^2 \times \dfrac{n}{n^{2/3}} = \sqrt[3]{n} \times 4\pi R^2$.

So the small balls together have $\sqrt[3]{n}$ times the surface area of the big ball. Eight small balls have $2$ times the area. Twenty-seven small balls have $3$ times the area. This is why sugar powder dissolves faster than a sugar cube. The powder has far more surface touching the water.

Practise with new numbers

Try it in Python

This program melts a cylinder and makes small balls. It prints the number of balls and compares the surface areas before and after.

Common mistakes

• Equating surface areas. The volume is the quantity that stays the same.
• Mixing units. A wire of radius $2\ \text{mm}$ has radius $0.2\ \text{cm}$.
• Forgetting that the number of pieces multiplies the small volume. With $n$ balls, write $n \times \frac{4}{3}\pi r^3$.
• Rounding too early. Keep $\pi$ in the equation. It often cancels from both sides.

Remember these

• When a solid is melted and recast, the volume stays the same.
• The number of small solids is the old volume divided by the volume of one.
• Surface area does not stay the same. $n$ equal balls from one big ball have $\sqrt[3]{n}$ times the area.
• Change all lengths to one unit before you equate volumes.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A cube of edge $6\ \text{cm}$ is melted and recast into a cuboid with a square base of side $3\ \text{cm}$. What is the height of the cuboid?

2. A ball of radius $3\ \text{cm}$ is melted and made into small balls of radius $1\ \text{cm}$. How many are made?

3. A wire of radius $0.1\ \text{cm}$ and length $50\ \text{cm}$ is melted and made into a ball. Find the volume of the ball in terms of $\pi$.

4. A solid is melted and made into $8$ equal balls. How does the total surface area compare with the first solid, if the first solid was one ball?

5. Why is the volume the same after melting, but not the surface area?