Volume conservation and applications

The single most powerful principle in this chapter is conservation of volume: when a solid is melted, recast, or poured from one container to another, the total volume of material does not change. The surface area usually does change , sometimes drastically , but the volume is preserved.

This principle turns countless word problems into one-line equations.

The strategy

1. Read the problem and identify the initial shape and its dimensions.
2. Identify the final shape and its unknown dimension.
3. Compute the initial volume.
4. Set it equal to the final volume.
5. Solve for the unknown.

Common scenarios

Melting a solid and recasting. A metal cone is melted and recast into a sphere; a metal sphere is melted and recast into smaller spheres; a metal cube is melted and recast into a cuboid. In all cases, $V_{\text{initial}} = V_{\text{final}}$ (or, for multiple smaller pieces, $V_{\text{initial}} = n \cdot V_{\text{small}}$).

Pouring water. Water from a cylinder is poured into a cone, or vice versa. The volume of water is unchanged. If the new container is partially filled, $V_{\text{water}} = V_{\text{cylinder up to the water level}}$.

Filling a tank. A tap with given flow rate fills a tank of known volume in time $T = V/r$.

Equating different shapes' volumes. A right circular cylinder and a sphere have equal volumes , find the relation between their dimensions.

Conversion ratios

Some often-used ratios in this chapter:

• A cone of radius $r$ and height $2r$ has volume $(1/3)\pi r^2(2r) = (2/3)\pi r^3$ , same as the hemisphere of radius $r$.
• A sphere of radius $r$ has volume $(4/3)\pi r^3$ , same as a cone of radius $r$ and height $4r$.
• Cylinder of radius $r$ and height $r$ has volume $\pi r^3$ , three times the cone of the same dimensions.

Worked examples

Example 1. A solid metallic sphere of radius $6$ cm is melted into a cylinder of radius $4$ cm. Find the height of the cylinder.

$V_{\text{sphere}} = (4/3)\pi(216) = 288\pi$.

$\pi(16) h = 288\pi \Rightarrow h = 18$ cm.

Example 2. A cone of radius $3$ and height $12$ is melted into a sphere. Find the radius.

$V_{\text{cone}} = (1/3)\pi(9)(12) = 36\pi$.

$(4/3)\pi R^3 = 36\pi \Rightarrow R^3 = 27 \Rightarrow R = 3$ cm.

Example 3. A solid sphere of radius $r$ is melted to form $n$ smaller solid spheres of radius $r/2$. Find $n$.

$(4/3)\pi r^3 = n \cdot (4/3)\pi(r/2)^3 = n \cdot (4/3)\pi r^3/8 \Rightarrow n = 8$.

Example 4. A cylindrical bucket of radius $7$ cm and height $40$ cm is filled with water. The water is then poured into a rectangular tank of base $11 \times 14$ cm$^2$. Find the height of water in the tank.

$V_{\text{bucket}} = (22/7)(49)(40) = 6160$ cm$^3$.

$V_{\text{tank}} = 11 \cdot 14 \cdot h \Rightarrow h = 6160/154 = 40$ cm.

Example 5. Water flows through a cylindrical pipe of radius $1$ cm at $7$ m/s. How many litres are delivered in $30$ minutes?

Cross-section area $= \pi(1)^2 = \pi$ cm$^2$.

Length per second $= 700$ cm.

Volume per second $= 700\pi$ cm$^3$.

Per minute $= 42000\pi$ cm$^3$. In $30$ minutes: $1\,260\,000 \pi \approx 3\,960\,000$ cm$^3 = 3960$ litres.

Example 6. A solid cuboid of dimensions $4 \times 3 \times 2$ cm is melted into a cube. Find the cube's side.

$V = 24$. $a^3 = 24 \Rightarrow a = \sqrt[3]{24} \approx 2.88$ cm.

Example 7. A cylindrical glass of radius $3$ cm is filled with water up to $5$ cm. A solid metallic sphere of radius $1.5$ cm is dropped into the glass. By how much does the water level rise (assuming no spillage)?

Volume displaced $= (4/3)\pi(1.5)^3 = (4/3)\pi(3.375) = 4.5\pi$ cm$^3$.

This volume occupies a height $\Delta h$ in the cylinder: $\pi(9) \Delta h = 4.5\pi \Rightarrow \Delta h = 0.5$ cm.

Example 8. A spherical ball of diameter $14$ cm is melted into a cylinder of base radius $14$ cm. Find the height of the cylinder.

$V = (4/3)(22/7)(7^3) = (4/3)(22/7)(343) = 4 \cdot 22 \cdot 343/(3 \cdot 7) = 1437.33$ cm$^3$.

$h = V/(\pi r^2) = 1437.33/(22/7 \cdot 196) = 1437.33/616 \approx 2.33$ cm.

Try it yourself

1. A sphere of radius $6$ is melted into a cylinder of radius $4$. Find the height.
2. A cone of radius $5$ and height $24$ is melted into a sphere. Find the radius.
3. A cube of side $10$ cm is melted into $n$ cubes of side $2$. Find $n$.
4. A cylindrical container of radius $3$ and height $14$ is filled with ice cream. It is moulded into cones of radius $1.5$ and height $4$. How many cones?
5. A cone of radius $9$ and height $12$ is melted into smaller cones of radius $3$ and height $4$. How many smaller cones?
6. A cylinder of radius $7$ and height $20$ is filled with water. Water is poured into a rectangular tank of $14 \times 11$ cm$^2$ base. Find the rise in water level.
7. Water flows through a cylindrical pipe of radius $2$ cm at $5$ m/s. How many litres in $30$ minutes?
8. A spherical lead shot of radius $5$ mm is melted into a cylindrical wire of radius $1$ mm. Find the length of the wire.
9. Three solid cubes of sides $3, 4, 5$ cm are melted into one cube. Find its side.
10. A cylinder of radius $14$ has water up to $5$ cm. A spherical iron ball of radius $7$ cm is dropped in. Find the rise in water level (assume no overflow).
11. A solid hemisphere of radius $r$ is melted into a cone of the same radius. Find the height.
12. A drum is in the form of a frustum with radii $30$ and $20$ cm and height $35$ cm. How many litres can it hold?

Pitfalls / Insight

(1) The key reflex: whenever you see "melted" or "recast" or "poured", equate volumes.

(2) Surface area is not conserved under melting. Computing the new surface area requires computing the new dimensions first.

(3) When pouring liquid, the volume of liquid is what's conserved, not the volume of the container. If the receiving container is larger than the liquid, the liquid takes a partial volume of that shape.

(4) Flow problems: Volume per unit time = (cross-section area) × (speed). So a pipe of radius $r$ delivering at speed $v$ pumps $\pi r^2 v$ volume per unit time.

(5) For partial filling problems (cone half-full, etc.), use similar triangles to relate the radius at the liquid surface to the height of the liquid.

(6) Always double-check units: convert m to cm or vice versa before multiplying.