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CHAPTER 3

Modelling & Estimation

AQA Mathematical Studies — Level 3 Certificate

Gulf Stream

Standard Form

Scaling

Subdividing

Fermi Estimation

Critical Evaluation

AQA MATHEMATICAL STUDIES

Level 3 Certificate

CHAPTER 3 — AQA MATHEMATICAL STUDIES

Chapter Overview & Learning Objectives

Prerequisites: estimating calculations · volume of cuboid/cylinder · index laws · standard form · percentages

SECTION 3.1

Modelling the Gulf Stream

Estimate volume of water flow using cross-sectional area, speed, and clearly stated modelling assumptions.

SECTION 3.2

Standard Form

Express and calculate with very large and very small numbers; multiply and divide in standard form.

SECTION 3.3

Estimation Technique 1: Scaling

Use a known reference quantity and scale proportionally to estimate unknown real-world values.

SECTION 3.4

Estimation Technique 2: Subdividing

Break complex regions or populations into smaller, manageable sub-parts and sum the estimates.

SECTION 3.5

Estimation Technique 3: Stating Assumptions

Identify, justify, and simplify assumptions to make real-world problems mathematically tractable.

SECTIONS 3.6 – 3.7

Facts & Formulae · Evaluating Models

Apply useful reference facts; critically assess model accuracy, limitations, and potential improvements.

Gulf Stream — warm current, Florida → Northern Europe

AQA MATHEMATICAL STUDIES · CHAPTER 3.1

Modelling the Gulf Stream

CONTEXT & ASSUMPTIONS

Warm current gives the UK its mild climate

Rectangular cross-section assumed

Speed ≈ 5 km/h (walking pace; accepted 6.4 km/h)

Depth D = 1 km , Width W = 100 km

KEY VALUES USED

Width = 100 km

Depth = 1 km

Speed = 5 km/h

1 km³ = 10⁹ m³

VOLUME CALCULATION

V = W × D × speed

=

100 × 1 × 5

=

500 km³/h

Convert: 500 km³/h = 500 × 10⁹ m³/h = 5 × 10¹¹ m³/h

THE MODELLING CYCLE

Represent mathematically

→

Use techniques

→

Interpret results

→

Compare with real data

→

Improve model

Gulf Stream Modelling

CHAPTER 3.1

AQA Advanced Maths — Chapter 3.1

Fermi Estimation: Ocean Current Modelling

EX

2

Example 1 — Gulf Stream

Volume of warm water past the British Isles per hour

ASSUMPTIONS

Rectangular cross-section

Depth D = 1 km , Width W = 100 km

Constant speed v = 5 km/h

STEP-BY-STEP CALCULATION

1

Volume = Depth × Width × Speed

V = 1 × 100 × 5 = 500 km³/h

2

Convert km³ to m³  (1 km³ = 10⁹ m³)

500 × 10⁹ = 5 × 10¹¹ m³/h

FINAL ANSWER

500 km³/h = 5 × 10¹¹ m³/h

500 × 10⁹ m³ per hour

Example 2 — California Current

Estimate water flow along the California coast per hour

ASSUMPTIONS

Rectangular cross-section

Depth D = 1 km , Width W = 40 km

Speed ≈ 5 km/h (same as Gulf Stream estimate)

STEP-BY-STEP CALCULATION

1

Volume = Depth × Width × Speed

V = 1 × 40 × 5 = 200 km³/h

2

Convert km³ to m³  (1 km³ = 10⁹ m³)

200 × 10⁹ = 2 × 10¹¹ m³/h

FINAL ANSWER

200 km³/h = 2 × 10¹¹ m³/h

200 × 10⁹ m³ per hour

AQA 3.2

Standard Form: Large & Small Numbers

a × 10 n

where 1 ≤ a < 10 and n is an integer

Multiply: multiply numbers, ADD powers

Divide: divide numbers, SUBTRACT powers

Small numbers use negative powers

MULTIPLYING IN STANDARD FORM

4 × 10 2 × 2 × 10 3 = ?

NUMBERS

4 × 2 = 8

POWERS

10 2 × 10 3 = 10 2+3 = 10 5

ANSWER

8 × 10 5

Add the powers when multiplying

DIVIDING IN STANDARD FORM

6 × 10 5 ÷ 2 × 10 3 = ?

NUMBERS

6 ÷ 2 = 3

POWERS

10 5 ÷ 10 3 = 10 5−3 = 10 2

ANSWER

3 × 10 2

Subtract the powers when dividing

NOT standard form:

15 × 10

 (15 > 10)  →  Rewrite as

1.5 × 10

. Always

check 1 ≤ a < 10!

NEGATIVE POWERS = SMALL NUMBERS

Negative power means the number is

less than 1

10 −7 = 0.000 000 1    (7 decimal places)

For Fermi estimation: round to

1 sig. fig.

first

KEY CONVERSIONS REFERENCE

10

11

Examples 1 & 2: Standard Form Calculations

Standard form: 1 ≤ a < 10, a × 10ⁿ

Example 1 — Distance to the Sun

AQA Ch 3.2 Example 2 · Multiplication of standard form

Given: speed of light = 299 792 458 m/s; time = 497 s. Round to 1 sig fig for Fermi estimation.

1

ROUND THE VALUES

Speed ≈ 3 × 10⁸ m/s  |  Time ≈ 5 × 10² s

2

APPLY: DISTANCE = SPEED × TIME

3 × 10⁸  ×  5 × 10²  =  15 × 10¹⁰

3

REWRITE IN STANDARD FORM (15 > 10, SO ADJUST)

15 × 10¹⁰  =  1.5 × 10¹¹ m

DISTANCE TO THE SUN

≈ 1.5 × 10¹¹ m

Standard form check: 1 ≤ 1.5 < 10 ✓  |  Powers added: 8 + 2 = 10, then adjusted to 11 ✓

Example 2 — Movies on a Hard Disc

AQA Ch 3.2 Example 3 · Division of standard form

Given: hard disc = 10¹³ bits; each movie ≈ 900 MB. Convert MB to bits first.

1

CONVERT MOVIE SIZE TO BITS

900 MB = 900 × 8 × 10⁶ = 7.2 × 10⁹ ≈ 7 × 10⁹ bits

2

DIVIDE: NUMBER OF MOVIES = DISC ÷ MOVIE SIZE

10¹³  ÷  7 × 10⁹  =  10⁴ / 7

3

EVALUATE THE RESULT

10 000 ÷ 7  ≈  1 500 movies

MOVIES STORED ON DISC

≈ 1 500 movies

Powers subtracted: 13 − 9 = 4  |  Result is a Fermi estimate — 1 sig fig is sufficient ✓

&

Exercise 3A — Questions

NO CALCULATOR (Q3)

Answer all questions. Show full working. Q3 must be completed without a calculator. Q4–Q6 require clear assumptions.

Note: Q4–Q6 are modelling and estimation problems — show all assumptions clearly. There is no single correct answer for Q4 and Q5.

Q1 — STANDARD FORM

Write in standard form

(a) 9 000    (b) 0.000 002

(c) 85 000 000    (d) 0.000 015

Q2 — CORRECT TO STANDARD FORM

Rewrite correctly in standard form

(a) 24 × 10³    (b) 360 × 10⁵

(c) 0.8 × 10³    (d) 0.03 × 10⁵

Q3 — MULTIPLY & DIVIDE (NO CALCULATOR)

Calculate, giving answers in standard form

(a) 2×10⁷ × 3×10⁴    (b) 6×10⁷ ÷ 3×10⁴

(c) (1.2×10⁷) × (1.2×10⁴)    (d) (4×10⁷) ÷ (8×10⁵)

(e) 6×10⁻² × 3×10⁴    (f) (6×10⁷) ÷ (3×10⁻⁴)

Q4 — MODELLING PROBLEM

Estimate the volume of your body

Model your body using cuboids and/or cylinders. State all dimensions assumed and show full working. Give your answer in cm³ and litres.

Q5 — ESTIMATION PROBLEM

Walking from Land's End to John O'Groats

Estimate how many days it would take to walk the full length of Great Britain. State your assumptions for distance and daily walking hours.

Q6 — PLANCK'S CONSTANT

Express in standard form

Planck's constant:

h = 0.000 000 000 000 000 000 000 000 000 000 000 662 6

Write h in the form a × 10ⁿ where 1 ≤ a < 10 .

Exercise 3A — Answers

✓ WORKED ANSWERS

Full worked answers for Q1–Q6. Green values indicate correct standard form — coefficient must satisfy 1 ≤ a < 10.

Key Rule: Standard form requires the coefficient a to satisfy 1 ≤ a < 10 . If a ≥ 10 or a < 1, adjust the power of 10 accordingly.

Q1 — CONVERTING TO STANDARD FORM

Write in standard form

(a) 9000 = 9 × 10³  |  (b) 0.000 002 = 2 × 10⁻⁶

(c) 85 000 000 = 8.5 × 10⁷  |  (d) 0.000 015 = 1.5 × 10⁻⁵

Q2 — CORRECTING TO STANDARD FORM

Rewrite correctly in standard form

(a) 24 × 10³ = 2.4 × 10⁴  |  (b) 360 × 10⁵ = 3.6 × 10⁷

(c) 0.8 × 10³ = 8 × 10²  |  (d) 0.03 × 10⁵ = 3 × 10³

Q3 — MULTIPLYING & DIVIDING (NO CALCULATOR)

Calculate without a calculator

(a) 2×10⁷ × 3×10⁴ = 6×10¹¹  |  (b) 6×10⁷ ÷ 3×10⁴ = 2×10³

(c) 1.2×10⁷ × 1.2×10⁴ = 1.44×10¹¹  |  (d) 4×10⁷ ÷ 8×10⁵ = 5×10¹

(e) 6×10⁻² × 3×10⁴ = 1.8×10³  |  (f) 6×10⁷ ÷ 3×10⁻⁴ = 2×10¹¹

Q4 — MODELLING: BODY VOLUME

Estimate volume of your body

Head ≈ sphere r=10 cm; torso ≈ cylinder r=15 cm, h=60 cm;

Arms ≈ cylinders r=4 cm, h=60 cm each; legs ≈ r=7 cm, h=90 cm each.

Total ≈ 70 000 cm³ = ≈ 70 litres

Q5 — MODELLING: LAND'S END TO JOHN O'GROATS

Estimate walking time

Distance ≈ 1400 km; walking speed ≈ 5 km/h; 8 hours/day.

Time = 1400 ÷ (5 × 8) = ≈ 35 days

Q6 — PLANCK'S CONSTANT

Express h in standard form

h = 0.000 000 000 000 000 000 000 000 000 000 000 662 6

= 6.626 × 10⁻³⁴ J·s

3.3 Estimation Technique 1: Scaling

Scaling: Estimate a quantity that can be multiplied or divided to reach the required quantity — choose something known or easier to estimate, then scale up or down.

Example 5 — UK Heartbeats/Year

Scale from pulse rate → national total

1

MEASURE PULSE

Count 20 beats in 15 seconds

20 ÷ 15 × 60 = 80 beats/min

2

SCALE PER PERSON PER YEAR

80 × 60 × 24 × 365 — round to simplify

≈ 100 × 50 × 20 × 400 = 4 × 10⁷ beats/year

3

SCALE TO UK POPULATION

Multiply by UK population ≈ 6 × 10⁷ people

6 × 10⁷ × 4 × 10⁷ = 24 × 10¹⁴

TOTAL UK HEARTBEATS/YEAR

2.4 × 10¹⁵ beats per year

Example 6 — Radius of the Earth

Scale from UK length → Earth's radius

1

KNOWN QUANTITY

Estimate the length of the UK (Land's End to John O'Groats)

UK length ≈ 1 000 km

2

VISUAL SCALING FROM GLOBE

From a globe, the Earth's radius looks about 6 × the UK length

radius ≈ 6 × 1 000 km

3

CHECK AGAINST ACTUAL VALUE

Actual radius of Earth = 6 371 km — very close!

estimate: 6 000 km ✓

ESTIMATED EARTH RADIUS

6 000 km (actual: 6 371 km)

3.4 Estimation Technique 2: Subdividing

Subdividing: Break a difficult estimation problem into smaller, more manageable geometric parts — then combine the results.

EXAMPLE 7

Land Area of the British Isles

GREAT BRITAIN

+

IRELAND

=

≈ 3×10⁵

km²

1

Approximate Great Britain as a triangle: height 1000 km , base 500 km

2

Area of GB = ½ × 500 × 1000 = 2.5 × 10⁵ km²

3

Approximate Ireland as a square: side 250 km

4

Area of Ireland = 250² ≈ 6 × 10⁴ km²

5

Total = 2.5×10⁵ + 6×10⁴ ≈ 3 × 10⁵ km²

Total land area of British Isles ≈ 3 × 10⁵ km²

EXAMPLE 8

Volume of Water in Earth's Oceans

Radius of Earth ≈ 6000 km

Ocean coverage ≈ 70% of surface

Mean depth ≈ 4 km

1

Radius of Earth ≈ 6000 km

2

Surface area = 4πr² ≈ 4 × 3 × 6000² ≈ 4 × 10⁸ km²

3

70% is ocean → ocean surface ≈ 3 × 10⁸ km²

4

Mean ocean depth ≈ 4 km

5

Volume = 3×10⁸ × 4 ≈ 10⁹ km³

Volume of Earth's oceans ≈ 10⁹ km³

Exercise 3B — Questions

CHAPTER 3.4

💡 All data needed is in the Reference Panel on the left — use Speed = Distance ÷ Time and the Gulf Stream facts to answer Q4–Q6.

📋 REFERENCE DATA — USE THESE TO ANSWER THE QUESTIONS

Pulse:

Count beats for 10 seconds, then × 6 = beats per minute

Heart rate:

Typical resting rate ≈ 60–80 beats per minute

Lifetime to age 18:

18 years × 365 days × 24 h × 60 min

England → New York:

Distance ≈ 5 500 km

American ships:

Journey time ≈ 20 days (mid-19th century)

Gulf Stream:

Warm current flowing NE from Florida to UK

Gulf Stream speed:

≈ 6 km/h (flows towards UK, not towards USA)

Gulf Stream length:

Florida to UK ≈ 7 000 km

Africa (N–S):

Estimate from globe — compare to known distances

Mercator maps:

Distort size near the poles — Greenland appears much larger than it really is

Speed formula:

Speed = Distance ÷ Time

Time formula:

Time = Distance ÷ Speed

Q1 — GLOBE ESTIMATION

Use a globe to estimate:

(a) Distance south to north of Africa

(b) Distance west to east of Africa at its widest point

(c) Length of the Gulf Stream from Florida to the UK

Q2 — SCALING

Count your pulse for 10 seconds. Estimate how many times your heart beats from birth to your 18th birthday.

Q3 — MAP DISTORTION

Daisy estimated Greenland's length from a map and found it very inaccurate.

(a) What might she not have noticed about the map?

(b) Would this make her estimate too small or too large?

(c) How could she improve her method?

Q4 — SPEED CALCULATION

American ships sailed England → New York in ≈ 20 days. Estimate their average speed in km/h. (Distance ≈ 5 500 km)

Q5 — GULF STREAM REASONING

British ships took significantly longer for the same journey.

(a) What could have affected their speed? (Hint: think about the Gulf Stream direction)

(b) How had American captains avoided this problem?

Q6 — FERMI ESTIMATE

Estimate how much longer British ships took. Use Gulf Stream speed ≈ 6 km/h and ship speed ≈ 11 km/h.

Exercise 3B — Answers

✓ WORKED SOLUTIONS

Full worked answers for all six questions in Exercise 3B — estimation using globes, Fermi techniques, and real-world geography.

Key Insight: Gulf Stream knowledge gave American captains a decisive navigational advantage — a real-world example of how estimation and geography intersect.

Q1 / GLOBE ESTIMATION

Africa Dimensions & Gulf Stream Length

(a) Africa north–south ≈ 8 000 km  |  (b) Africa east–west ≈ 7 500 km  |  (c) Gulf Stream (Florida → UK) ≈ 7 000 km

Q2 / FERMI ESTIMATE

Heartbeats from Birth to Age 18

80 × 60 × 24 × 365 × 18 ≈ 7.6 × 10⁸ beats. Approximately 760 million heartbeats by your 18th birthday.

Q3 / MAP DISTORTION

Greenland & Mercator Projection

(a) Mercator projection greatly exaggerates land area near the poles.  (b) Estimate would be too large .  (c) Use a globe or an equal-area projection map.

Q4 / SPEED CALCULATION

American Ships: England to New York

Distance ≈ 5 500 km . Time = 20 days × 24 h = 480 h . Average speed = 5 500 ÷ 480 ≈ 11 km/h .

Q5 / GULF STREAM

Why British Ships Were Slower

(a) British ships sailed against the Gulf Stream current, adding resistance.  (b) American captains knew about the Gulf Stream and used it to speed their return journey.

Q6 / FERMI ESTIMATE

How Much Longer Did British Ships Take?

Gulf Stream speed ≈ 6 km/h resistance added to journey. This makes the trip roughly 50% longer → approximately 10 extra days compared to American ships.

3.5 Estimation Technique 3: Stating Assumptions

EXAMPLE 9

Key Principle: State assumptions clearly so they can be checked and improved later — even rough assumptions can give surprisingly accurate results.

ASSUMPTIONS

CALCULATION STEPS

Step 1

Volume for fish

= 10¹⁸ × (20 ÷ 4000)

= 5 × 10¹⁵ m³

Step 2

Number of fish

= 5 × 10¹⁵ ÷ 10³

= 5 × 10¹²

Step 3

Final answer

5 × 10¹² fish

≈ 5 trillion fish

OUR ESTIMATE

5 × 10¹²

≈ 5 trillion fish

Finding Nemo claimed there are 3.7 trillion fish in the ocean — remarkably close to our estimate!

OCEAN VOLUME

10⁹ km³

= 10¹⁸ m³

AVG. DEPTH

4 km

= 4000 m total

FISH ZONE

Top 20 m

Most fish live near surface

TERRITORY / FISH

10 m cube

= 10³ m³ per fish

Earth's oceans — how many fish?

Example 1 & 2: Stating Assumptions

AQA Chapter 3.5

Key Principle: Always state your assumptions clearly before calculating — this allows others (and you) to review, challenge, and improve them later.

&

Example 1: Fish in Earth's Oceans

How many fish are there? (Example 9 in book)

ASSUMPTIONS

Volume of oceans = 10⁹ km³ = 10¹⁸ m³

Most fish live in top 20 m below surface

Each fish has territory = cube of side 10 m = 10³ m³

CALCULATION

1

Volume for fish = 10¹⁸ × (20 ÷ 4000) = 5 × 10¹⁵ m³

2

Number of fish = 5 × 10¹⁵ ÷ 10³ = 5 × 10¹²

ANSWER

5 × 10¹² ≈ 5 trillion fish

Finding Nemo estimated 3.7 trillion — remarkably close!

Key: state each assumption clearly so it can be checked and improved later

Example 2: Shanghai Maglev Journey

How long to travel the length of Great Britain? (Example 10)

ASSUMPTIONS

Shanghai Maglev speed = 430 km/h

Length of Great Britain ≈ 1000 km

Approximate 430 ≈ 400 for easier mental arithmetic

CALCULATION

1

Formula: Time = Distance ÷ Speed

2

Time = 1000 ÷ 430 ≈ 1000 ÷ 400 = 2.5 hours

ANSWER

≈ 2.5 hours

Rounding 430 → 400 makes division straightforward

Word formula: Speed = Distance ÷ Time  →  Time = Distance ÷ Speed

3.6

Useful Facts & Formulae

Human Body

▸

80 kg

— typical weight

▸

1.7 m

— typical height

▸

75 yr

— lifetime

▸

80 bpm

— heart rate

▸

5 L

— blood volume

▸

10 m/s

sprint · 3 mph walk

Earth

▸

6 000 km

— radius

▸

1 rotation

per day

▸

1 orbit

per year (365 days)

▸

10⁹ km³

— ocean volume

▸

4 km

— avg ocean depth

Water & Density

▸

1 g/cm³

— density of water

▸

1 litre

= 1 000 cm³

▸

1 m³

= 1 000 litres

▸

Density = Mass ÷ Volume

Data Storage

▸

1 bit

= binary digit (0 or 1)

▸

1 byte

= 8 bits

▸

1 KB

= 10³ bytes

▸

1 MB

= 10⁶ bytes

▸

1 GB

= 10⁹ bytes

Equivalences

▸

1 mile

≈ 1 600 m

▸

1 foot

≈ 30 cm

▸

1 pound

≈ 500 g

▸

1 year

≈ 3.15 × 10⁷ s

▸

1 hour

= 3 600 s

Key Formulae

▸

V = l³

— cube volume

▸

A = 4πr²

— sphere surface

▸

V = (4/3)πr³

— sphere volume

▸

T = D ÷ S

— time = dist ÷ speed

▸

A = πr²

— circle area

AQA Advanced Maths — Chapter 3.6 Reference

CHAPTER 3.6 — WORKED EXAMPLES

Example 1 & 2: Using Facts & Formulae

EXAMPLE 1 (BOOK: EX. 11)

Earth's Density

GIVEN

Mass of Earth = 6 × 10²⁷ g

Volume of Earth = 1.1 × 10²⁷ cm³

1

Write the formula

Density = Mass ÷ Volume

2

Substitute values

= (6 × 10²⁷) ÷ (1.1 × 10²⁷)

3

Divide coefficients; subtract powers: 27 − 27 = 0

= (6 ÷ 1.1) × 10⁰ ≈ 5.5 × 1

Density ≈ 5.5 g/cm³

Earth is the most dense planet in the Solar System

EXAMPLE 2 (BOOK: EX. 4)

Water Molecules in Your Body

GIVEN

Body mass = 70 kg = 7 × 10¹ kg

Mass of H₂O molecule = 3 × 10⁻²⁶ kg

1

Write the formula

Number = Body mass ÷ Molecule mass

2

Substitute values

= (7 × 10¹) ÷ (3 × 10⁻²⁶)

3

Divide coefficients; subtract powers: 1 − (−26) = 27

= (7 ÷ 3) × 10²⁷ ≈ 2.3 × 10²⁷

≈ 2 × 10²⁷ molecules

That's 2 octillion water molecules in a 70 kg body

÷

AQA CHAPTER 3.7 — MODELLING CYCLE: STEP 4

Critically Evaluating Models — Light Through History

55 BC

Lucretius

Light = particles travelling to the eye (Ancient Greece)

Late 1600s

Newton (corpuscles) vs Huygens (waves) —competing models emerge

1700s

Newton's corpuscular theory prevails — particle model dominates

1800s

Maxwell's wave theory becomes accepted — light travels as electromagnetic waves

Early 1900s

Einstein — photoelectric effect proves light is photons (particles again)

21st Century

Wave–particle duality —light exhibits both properties simultaneously

Key Lesson: Models are refined over time — a good model that is later improved is still valuable. It paves the way for further progress. The modelling cycle: Represent → Use techniques → Interpret → Compare with data → Improve.

EARLY / SUPERSEDED

ESTABLISHED MODEL

CURRENT UNDERSTANDING

Consolidation Exercise 3 — Questions

CHAPTER 3 · AQA MATHEMATICAL STUDIES

Apply your modelling and estimation skills to the following six questions. State all assumptions clearly and show your working.

Technique reminder: For each question, identify whether you are using scaling , subdividing , or known facts and formulae — and state your assumptions explicitly.

Q1 · ESTIMATION

Fish in the Oceans

How many fish are there in the Earth's oceans? State all assumptions you make and explain your estimation technique.

Q4 · REAL-WORLD CONTEXT

Grace Madeje's Water Journey

Grace lives 11 km from the nearest water source. Estimate how many hours per day she spends collecting water.

Q2 · STANDARD FORM

Antarctic Ice Melt — Sea Level Rise

If all Antarctic ice melts, by how much will the oceans rise? Ice cap area ≈ 14 ×10⁶ km², thickness ≈ 2 km, ocean area ≈ 3 × 10⁸ km².

Q5 · FORMULAE

Blood Pumped in a Lifetime

Estimate the total volume of blood your heart pumps in a lifetime. Blood ≈ 7% of body volume; all blood passes through the heart every 1 minute.

Q3 · SCALING

Access to Safe Water

How many people worldwide do not have access to safe water? Roughly 1 in 9 people lack safe water access.

Q6 · STANDARD FORM

Voyager 1 to Alpha Centauri

Voyager 1 travels at 6.1 × 10⁴ km/h . Distance to Alpha Centauri = 2.4 light years . How many years to travel there?

Full worked answers for all six questions. Key steps shown for each calculation — all use standard form and estimation techniques from Chapter 3.

Key Takeaway: All six answers apply standard form arithmetic — multiply/divide powers of 10, state assumptions clearly, and interpret results in context.

Q1 / FISH IN THE OCEANS

≈ 5 × 10¹² fish (5 trillion)

See Example 9 for full working. Scale from known fish densities across ocean zones to estimate total global population.

Q4 / GRACE'S WATER COLLECTION

≈ 5.5 hours per day

11 km each way → 22 km round trip. Walking speed ≈ 4 km/h → 22 ÷ 4 ≈ 5.5 hours/day collecting water.

Q2 / ANTARCTIC ICE MELT — SEA LEVEL RISE

Sea level rise ≈ 90 m

Ice volume = 14×10⁶ × 2 = 2.8×10⁷ km³. Rise = 2.8×10⁷ ÷ 3×10⁸ ≈ 0.09 km = 90 m .

Q5 / BLOOD PUMPED IN A LIFETIME

≈ 2 × 10⁸ litres

Blood ≈ 5 L, pumped every minute. 5 × 60 × 24 × 365 × 75 ≈ 2 × 10⁸ litres over a 75-year lifetime.

Q3 / PEOPLE WITHOUT SAFE WATER

≈ 9 × 10⁸ people (900 million)

World population ≈ 8×10⁹. 1/9 × 8×10⁹ ≈ 9×10⁸ . That is about 13 times the UK population (6.7×10⁷).

Q6 / VOYAGER 1 TO ALPHA CENTAURI

≈ 42,000 years

1 light year ≈ 9.5×10¹² km. Distance = 2.4×9.5×10¹² ≈ 2.3×10¹³ km. Time = 2.3×10¹³ ÷ 6.1×10⁴ ÷ 8760 ≈ 42,000 years .

Consolidation Exercise 3 — Answers

✓ WORKED SOLUTIONS

Key Formulae & Methods Summary

Chapter 3 · Sections 3.1–3.7

STANDARD FORM

a × 10ⁿ  where 1 ≤ a < 10

Expresses very large or very small numbers in compact scientific notation.

MULTIPLYING STD FORM

(a × 10ᵐ) × (b × 10ⁿ)

= (a × b) × 10^(m+n)

Multiply the coefficients; add the powers of 10.

DIVIDING STD FORM

(a × 10ᵐ) ÷ (b × 10ⁿ)

= (a ÷ b) × 10^(m−n)

Divide the coefficients; subtract the powers of 10.

VOLUME FORMULAE

Cuboid: V = l × w × h

Cylinder: V = πr²h

Used to model real-world volumes such as ocean currents and ice caps.

SURFACE AREA — SPHERE

SA = 4πr²

V = (4/3)πr³

Surface area and volume of a sphere; key for planetary and biological models.

SPEED & DENSITY

Speed = Distance ÷ Time

Density = Mass ÷ Volume

Applied to Voyager 1 travel time and Earth's density calculations.

ESTIMATION TECHNIQUES

① Scaling — known → unknown

② Subdividing — break into parts

③ State all assumptions clearly. Fermi estimation builds on these three strategies.

MODELLING CYCLE

Represent → Techniques

→ Interpret → Compare →Improve

Critically evaluate and refine models; good models pave the way for further progress.

Interactive Quiz — Modelling & Estimation

CHAPTER 3

1

2

3

4

5

Question 1 of 5

TIME

29

Reveal Answer

Next

Restart

QUESTION 1 — STANDARD FORM

Write 85 000 000 in standard form.

Remember: standard form is a × 10ⁿ where 1 ≤ a < 10

Prev

CHAPTER SUMMARY

TOPICS MASTERED

Gulf Stream modelling — volume flow, real-world geometric approximation

Standard form — multiply & divide powers of 10 without a calculator

Three Fermi techniques — scaling, subdividing, and stating assumptions clearly

Critical evaluation — refining models using the four-step modelling cycle

NEXT STEPS

Practise past AQA exam questions on standard form and estimation

Apply the modelling cycle to real-world problems of your choice

Review the key formulae card — volume, surface area, speed, density

Well done! A model that is later improved is still valuable — it paves the way for further progress. Keep thinking mathematically.

AQA MATHEMATICAL STUDIES

Chapter 3

Complete!

Modelling & Estimation — from the Gulf Stream to the stars. You've mastered the tools of mathematical thinking.

Gulf Stream

Standard Form

Fermi Estimation

Critical Evaluation

Chapter 3 Quiz

MODELLING & ESTIMATION

QUESTION 1 OF 5

Write 85 000 000 in standard form.

29

Reveal Answer