Mathematics • Year 10 • Unit 3 • Lesson 11

Scale Factors in the Real World

Apply the k, k², k³ rules to real Australian contexts — model cars, photo enlargements, scale plans for a Newcastle home, and the surface area of two similar water tanks at a Hunter Valley vineyard. Every problem starts with the linear scale factor, then uses the correct power for the quantity asked.

Apply · Real-World Maths

1. Word problems

For each problem: (i) identify the linear scale factor k, (ii) decide whether to use k, k² or k³, (iii) calculate. Show working.

1.1 — Model car. A 1:24 scale model is built of a real Holden that is 4.8 m long. The real car's bonnet (looking from above) has an area of 2.4 m².

(a) How long is the model, in cm?
(b) What is the model bonnet's area, in cm²? (Hint: it is not simply 2.4 / 24.) 4 marks

Stuck? "1:24 scale" means k = 1/24 (model ÷ real). Lengths scale by k but area scales by k².

1.2 — Photo enlargement. A school photo printed at 10 cm × 15 cm is enlarged at the photo lab so every dimension is tripled.

(a) State the new dimensions.
(b) Find the new area, and state by what factor the area has increased. 3 marks

Stuck? "Tripled" gives k = 3. New area = original × k².

1.3 — Architect's scale plan. An architect draws a plan of a Newcastle living room at a scale of 1:50. On the plan the room measures 8 cm by 6 cm.

(a) Find the actual dimensions of the room in metres.
(b) Find the actual floor area in m². 3 marks

Stuck? 1 cm on the plan represents 50 cm in real life. Convert to metres before computing area.

1.4 — Two water tanks. Two similar cylindrical water tanks at a Hunter Valley vineyard have heights 2 m and 5 m. The smaller tank holds 1.6 kL.

(a) Find the linear scale factor (large / small).
(b) Find how many kilolitres the larger tank holds. 3 marks

Stuck? Capacity is a volume, so use k³.

1.5 — Working backwards from area. Two similar gardens at Centennial Park have areas of 100 m² and 225 m². The perimeter of the smaller garden is 40 m. Find the perimeter of the larger garden. 3 marks

Stuck? Perimeter is a length, so it scales by k (not k²). Find k from the area ratio first: k = √(area ratio).

2. Explain your thinking

This question is about communication, not just numbers. Use full sentences. 4 marks

2.1 A friend says: "If a copier doubles every length on a page, the photocopied area also doubles." Using the language from Lesson 11 (scale factor, k², two dimensions), explain in 4-6 sentences (i) why the friend is wrong, (ii) what the area actually does, and (iii) why a concrete number example makes this obvious. Reference a specific original-and-image area pair in your explanation.

Stuck? Revisit lesson § "Common Pitfalls" — "Forgetting that area scales by k²" is called out explicitly.

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Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — 1:24 model car

k = 1/24.
(a) Model length = 4.8 m ÷ 24 = 0.2 m = 20 cm.
(b) Area scale factor = k² = (1/24)² = 1/576.
Model bonnet area = 2.4 m² ÷ 576 = 2.4 × 10000 cm² ÷ 576 = 24000 ÷ 576 ≈ 41.7 cm².
Note: 2.4 m² = 24000 cm². The naive answer 2.4 / 24 = 0.1 m² would be 1000 cm² — far too big.

1.2 — Photo enlargement, k = 3

(a) New dimensions = (10 × 3) cm × (15 × 3) cm = 30 cm × 45 cm.
(b) Original area = 10 × 15 = 150 cm². New area = 30 × 45 = 1350 cm². The area has increased by a factor of 1350 / 150 = 9 = k².

1.3 — Architect's plan, scale 1:50

(a) 1 cm on the plan = 50 cm = 0.5 m in real life.
Actual length = 8 × 0.5 = 4 m. Actual width = 6 × 0.5 = 3 m.
(b) Actual area = 4 × 3 = 12 m².
Check using k²: plan area = 48 cm². Linear scale (real/plan) = 50, so area scale = 2500. Actual area = 48 × 2500 = 120000 cm² = 12 m² ✓.

1.4 — Two water tanks

(a) k = 5 / 2 = 2.5.
(b) Volume scale factor = k³ = 2.5³ = 15.625.
Larger tank = 1.6 × 15.625 = 25 kL.

1.5 — Gardens of area 100 m² and 225 m²

Area ratio = 225 / 100 = 9/4.
Linear scale factor k = √(9/4) = 3/2 = 1.5.
Larger perimeter = 40 × 1.5 = 60 m.
Perimeter scales by k, never k². Going via the area ratio is the only way when you are not given lengths directly.

2.1 — Explain your thinking (sample response)

My friend is wrong because lengths and areas do not scale by the same factor. If every length is multiplied by the linear scale factor k = 2, the area is multiplied by k² = 4, not 2. Area depends on two dimensions: both the length and the width are doubled, so the area becomes 2 × 2 = 4 times larger. A concrete example makes this obvious: an original 10 cm × 15 cm photo has area 150 cm², but after doubling, the new 20 cm × 30 cm photo has area 600 cm² — exactly four times the original, not double it. The rule is "length × k, area × k², volume × k³".

Marking: 1 for stating area scales by k², 1 for the "two dimensions" reason, 1 for the concrete number example with original and new area, 1 for the summary rule.