Mathematics • Year 9 • Unit 4 • Lesson 18
Measurement in the Real World
Use surface area, volume and scale factors in authentic Y9 contexts: water tanks, packaging, architecture, swimming pools, and 3D printing. Then explain why doubling all the lengths of a tank multiplies its capacity by 8.
1. Word problems
Choose the correct formula. Watch unit conversions: 1 m³ = 1 000 L, 1 m³ = 1 000 000 cm³.
1.1 — Cylindrical water tank. A rainwater tank is a cylinder with radius 1.5 m and height 3 m.
(a) Find the tank's capacity in m³ to 2 d.p.
(b) Convert this to litres.
(c) If a household uses 200 L of water per day, how many days will a full tank last? 4 marks
1.2 — Packaging design. A cardboard box is a rectangular prism 30 cm × 20 cm × 15 cm.
(a) Find the volume in cm³.
(b) Find the total surface area (the total cardboard needed) in cm².
(c) A new design doubles every dimension. How many times more cardboard, and how many times more volume? 4 marks
1.3 — Scale model of a building. An architect builds a 1 : 50 scale model of an office tower. The real tower will use 24 000 m³ of concrete.
(a) Find the volume of the model in m³.
(b) Convert to cm³.
(c) If the real tower's footprint (floor area) is 400 m², find the model footprint in cm². 4 marks
1.4 — Swimming pool refill. A rectangular pool is 25 m long, 10 m wide, with a uniform depth of 1.5 m.
(a) Find the volume of water needed to fill it in m³.
(b) Convert to litres.
(c) Refill via a hose at 30 L/min. How many hours will it take to fill from empty? (Round to the nearest hour.) 4 marks
1.5 — 3D-printed scale figures. A 3D printer makes a small figurine 8 cm tall using 30 g of filament. A customer orders the same model scaled up so it is 20 cm tall.
(a) Find the length scale factor k from the small to the larger figurine.
(b) Assuming the filament use scales like volume, how much filament does the large figurine use?
(c) Filament costs $0.05 per gram. What's the extra cost for the larger order, compared with the small one? 4 marks
2. Explain your thinking
Use full sentences. 4 marks
2.1 A family wants a new rainwater tank "double the size" of their old one. They assume that means buying a tank with double the height and double the radius. Explain in your own words:
(i) What capacity (in m³) the new tank will actually hold compared to the old one.
(ii) The lesson rule that explains this (one sentence).
(iii) What they should ask the supplier for if they really want "exactly double the capacity" (one practical suggestion).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Rainwater tank
(a) V = π × 1.5² × 3 = π × 2.25 × 3 = 6.75π ≈ 21.21 m³.
(b) 21.21 × 1 000 = 21 210 L.
(c) 21 210 ÷ 200 ≈ 106 days.
1.2 — Cardboard box
(a) V = 30 × 20 × 15 = 9 000 cm³.
(b) SA = 2(30×20 + 30×15 + 20×15) = 2(600 + 450 + 300) = 2 × 1 350 = 2 700 cm².
(c) Cardboard scales by k² = 4× (10 800 cm²); volume scales by k³ = 8× (72 000 cm³).
1.3 — Scale model 1:50
Volume scale = 50³ = 125 000.
(a) Model volume = 24 000 ÷ 125 000 = 0.192 m³.
(b) 0.192 × 1 000 000 = 192 000 cm³.
(c) Area scale = 50² = 2 500. Model footprint = 400 ÷ 2 500 = 0.16 m² = 0.16 × 10 000 = 1 600 cm².
1.4 — Swimming pool
(a) V = 25 × 10 × 1.5 = 375 m³.
(b) 375 × 1 000 = 375 000 L.
(c) Time = 375 000 ÷ 30 = 12 500 min ÷ 60 ≈ 208 hours ≈ 9 days.
1.5 — 3D-printed figurines
(a) k = 20 ÷ 8 = 2.5.
(b) Filament for large = k³ × 30 = 2.5³ × 30 = 15.625 × 30 = 468.75 g.
(c) Extra cost = (468.75 − 30) × $0.05 = 438.75 × 0.05 = $21.94.
2.1 — Explain your thinking (sample response)
(i) If they double both the radius AND the height, every length is scaled by k = 2, and volume scales by k³ = 8×. So the new tank holds eight times the old capacity, not two times. (ii) The lesson rule: "if lengths scale by k, volume scales by k³" — doubling every dimension multiplies the volume by 2³ = 8. (iii) To get exactly double the capacity, they should ask for a tank with the same radius but double the height, or any combination where the new V is exactly 2× the old V (e.g. radius × √2 with same height). They should specify the capacity in litres, not "double the size".
Marking: 1 mark for the correct "8×" answer; 1 for naming the k³ rule; 1 for a workable practical suggestion; 1 for full-sentence communication.