CDS Mathematics · Mensuration 3D

Water — Immersion, Flow & Rainfall

A solid dropped into water raises the level by its own volume; water flowing through a pipe or a river, or falling as rain, is a volume that must be converted between cm, m, litres and hectares.

All Mensuration 3D notes CDS Mathematics strategy

Why this matters

Easy to set up and easy to get wrong: the physics is one line, and the marks go on units. Learn the four conversions below and the page becomes arithmetic.

Concept 1 of 2: Immersion raises the level

A solid fully under water pushes aside exactly its own volume. In a cylindrical vessel that volume spreads over the circular base, so the rise is the solid's volume divided by the base area.

Definition

Rise $= \dfrac{\text{volume immersed}}{\pi R^2}$ in a cylinder of radius $R$ .
Many balls: multiply one ball's volume by their number.
Water left after a solid is lowered into a full vessel $=$ vessel $-$ solid.
Pouring from one vessel to another keeps the volume: new depth $= \dfrac{\text{volume}}{\text{new base area}}$ .

Rise in a cylinder

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

Worked example

A sphere of radius

6

cm is dropped into a cylinder of radius

8

cm partly filled with water. By how much does the level rise?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

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

Example 1 · Mensuration 3D · Water — Immersion, Flow and Rainfall

A sphere of diameter 6 cm is dropped into a cylindrical vessel partly filled with water. The radius of the vessel is 6 cm. If the sphere is completely submerged in water, then by how much will the surface level of water be raised?

Drill 6 more on immersion raises the level

Concept 2 of 2: Rainfall and flow — watch the units

Rain is a thin slab: area times depth. Flow is a moving column: cross-section times speed times time. The only difficulty is units, so convert everything to metres (or everything to centimetres) before multiplying.

Definition

$1$ m $^3 = 1000$ litres; $1$ litre $= 1000$ cm $^3$ ; $1$ kL $= 1$ m $^3$ .
$1$ hectare $= 10000$ m $^2$ .
Water weighs $1$ g per cm $^3$ , i.e. $1$ tonne per m $^3$ .
Rainfall volume $=$ area $\times$ depth; flow per unit time $=$ cross-section $\times$ speed.
A tank losing $V$ of water drops by $\dfrac{V}{\text{base area}}$ .

Flow

\text{volume per unit time} = \text{cross-section} \times \text{speed}

Worked example

A canal

5

m wide and

2

m deep flows at

3

km/h. How many litres pass a point in a minute?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2020 · CDS (II) 2020 — Elementary Mathematics · Q22Moderate

Example 2 · Mensuration 3D · Water — Immersion, Flow and Rainfall

A river 3 m deep and 40 m wide is flowing at the rate of 2 km/hr and falls into the sea. What is the amount of water in litres that will fall into the sea from this river in a minute ?

Litres, cubic metres and cubic centimetres differ by thousands

The options on these questions are usually the same digits with different numbers of zeros. Write the unit next to every number until the last line.

Drill 9 more on rainfall and flow — watch the units

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)

Immersion raises the level
Rise in a cylinder
$\text{rise} = \frac{V_{\text{solid}}}{\pi R^2}$
Rainfall and flow — watch the units
Flow
$\text{volume per unit time} = \text{cross-section} \times \text{speed}$

Watch out for (1)

Litres, cubic metres and cubic centimetres differ by thousands
→ Rainfall and flow — watch the units

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.

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