Mathematics • Year 10 • Unit 3 • Lesson 13
Scale, Maps and Models — Real-World Ratios
Apply k, k² and k³ to map scales, models and similar tanks in Australian contexts: a topographic map of the Blue Mountains, a 1:50 architect's plan of a Sydney apartment, two similar grain silos, and a 1:18 die-cast Holden ute. Decide which power of k to use before reaching for the calculator.
1. Word problems
For each problem: state the linear scale factor k, decide whether to use k, k², or k³, and then calculate.
1.1 — Blue Mountains topographic map. A walking map is drawn to a scale of 1:25,000. On the map, a national park reserve is shown as a rectangle 6 cm by 4 cm.
(a) Find the actual dimensions, in km.
(b) Find the actual area, in km². 3 marks
1.2 — Architect's apartment plan. A floor plan of a Sydney apartment is drawn at 1:50. On the plan, the lounge room is 8 cm × 6 cm and the bedroom is 6 cm × 5 cm.
(a) Find the actual floor area of the lounge in m².
(b) Find the actual floor area of the bedroom in m².
(c) State the area scale factor (real / plan) and verify your lounge answer using it. 4 marks
1.3 — Two similar grain silos. Two similar cylindrical grain silos in the Riverina have heights 4 m and 10 m. The smaller silo holds 32 tonnes of wheat when full.
(a) Find the linear scale factor.
(b) Find how much wheat the larger silo holds. 3 marks
1.4 — Die-cast Holden ute, 1:18. A 1:18 die-cast model of a Holden ute is sold in a souvenir shop. The real ute is 5.4 m long, has a side-window area of 0.36 m², and weighs 1620 kg.
(a) Find the model length in cm.
(b) Find the model window area in cm².
(c) If the model is made of the same density of material, what would its weight be in g? 4 marks
1.5 — Working backwards. Two similar paddocks on a Coffs Harbour farm have areas of 1,600 m² and 3,600 m². The smaller paddock has a perimeter of 160 m. Find the perimeter of the larger paddock. 3 marks
2. Explain your thinking
This question is about communication, not just numbers. Use full sentences. 4 marks
2.1 A friend says: "The map says 1:50,000. So an area on the map of 12 cm² represents 12 × 50,000 = 600,000 cm² in real life." Using the language of Lesson 13 (linear scale factor, area scale factor, k², two dimensions), explain in 4-6 sentences (i) why this is wrong, (ii) what the correct multiplier for area is, and (iii) compute the correct real-world area for the 12 cm² map area. State your final answer in m² or km².
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Topographic map 1:25,000
1 cm on map = 25,000 cm = 0.25 km.
(a) Real length = 6 × 0.25 = 1.5 km. Real width = 4 × 0.25 = 1 km.
(b) Actual area = 1.5 × 1 = 1.5 km².
1.2 — Architect's plan 1:50
1 cm on plan = 0.5 m real.
(a) Lounge real dimensions = 4 m × 3 m. Area = 12 m².
(b) Bedroom real dimensions = 3 m × 2.5 m. Area = 7.5 m².
(c) Area scale factor (real/plan) = 50² = 2500. Plan lounge area = 48 cm². Real area in cm² = 48 × 2500 = 120,000 cm² = 12 m² ✓.
1.3 — Two grain silos
(a) k = 10 / 4 = 2.5.
(b) k³ = 2.5³ = 15.625.
Larger silo capacity = 32 × 15.625 = 500 tonnes.
1.4 — Die-cast Holden ute 1:18
k = 1/18.
(a) Model length = 5.4 m ÷ 18 = 0.3 m = 30 cm.
(b) Window area scale factor = k² = 1/324.
Model area = 0.36 m² ÷ 324 = 0.36 × 10000 cm² ÷ 324 = 3600 ÷ 324 ≈ 11.1 cm².
(c) Weight scale factor = k³ = 1/5832.
Model weight = 1620 kg ÷ 5832 ≈ 0.2778 kg ≈ 278 g.
1.5 — Two similar paddocks
Area ratio = 3600 / 1600 = 9/4. k = √(9/4) = 3/2.
Larger perimeter = 160 × (3/2) = 240 m.
Perimeter is a length, so it scales by k, not k².
2.1 — Explain your thinking (sample response)
My friend is wrong: a scale of 1:50,000 means each linear dimension is multiplied by 50,000, not each area. Area depends on two dimensions, so when both length and width are scaled by 50,000, the area is scaled by the area scale factor k² = 50,000² = 2.5 × 10⁹. The correct real-world area for a 12 cm² rectangle on the map is therefore 12 × 2.5 × 10⁹ = 3 × 10¹⁰ cm², which converts to 3 km² (since 1 km² = 10¹⁰ cm²). The friend's answer of 600,000 cm² (0.06 m²) is tiny — barely the size of a footprint — which is a useful sanity-check that something has gone wrong.
Marking: 1 for spotting that area uses k², 1 for stating k² = 2.5 × 10⁹, 1 for the correct numerical area, 1 for converting to a sensible unit (m² or km²) and a sanity-check sentence.