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.

How did this worksheet feel?

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.