Mathematics • Year 10 • Unit 1 • Lesson 17

Curved Surfaces in the Real World

Apply cylinder, cone and sphere surface-area formulas to real Australian situations — insulating hot-water cylinders, painting water tanks, costing waterproof fabric for tents, and wrapping a spherical ornament. Then explain your method in your own words.

Apply · Real-World Maths

1. Word problems

Each problem uses a curved-surface formula from Lesson 17: 2πrh (cylinder side), πrl (cone side with slant l = √(r²+h²)), or 4πr² (sphere). Round decimal answers to the precision asked for, and show every step.

1.1 — Insulating a hot-water cylinder. A standard Australian 250-litre hot-water tank is a closed cylinder with radius 27 cm and height 145 cm. Foam insulation costs $32 per m².

(a) Calculate the total surface area to be insulated (curved side + 2 ends), in m² to 2 d.p.
(b) Calculate the cost of the insulation. 3 marks

Stuck? Convert cm to m first (r = 0.27 m, h = 1.45 m), then use TSA = 2πrh + 2πr².

1.2 — Painting a rural water tank. A closed cylindrical rainwater tank on a farm has diameter 1.4 m and height 2.1 m. The owner wants to paint it; 1 litre of paint covers 8 m².

(a) Calculate the total external surface area of the tank.
(b) How many full litres of paint should the owner buy for one coat? 3 marks

Stuck? Divide the diameter by 2 first to get r = 0.7 m. Paint is sold in whole litres — round up.

1.3 — Waterproof fabric for a conical tent. A tipi-style conical tent has base diameter 3 m and perpendicular height 2 m. Waterproof fabric costs $28 per m² and only the curved surface (not the ground) is covered.

(a) Find the slant height of the tent.
(b) Find the curved surface area.
(c) Find the cost of the fabric. 4 marks

Stuck? r = 1.5 m. Slant l = √(1.5² + 2²) = √6.25 = 2.5 m. CSA = πrl, not πr².

1.4 — Spherical ornament gift wrap. A spherical decorative ornament has diameter 18 cm. The gift wrap shop charges $0.04 per cm² and adds 20% extra for overlap and folds.

(a) Calculate the surface area of the ornament.
(b) Calculate the area of gift wrap needed including the 20% extra.
(c) Calculate the cost of wrapping. 3 marks

Stuck? Radius = 18 ÷ 2 = 9 cm. SA = 4π(9)² = 324π. Multiply by 1.20 for the extra, then by $0.04.

1.5 — Composite ice-cream cone. An ice-cream consists of a cone of radius 3 cm and perpendicular height 10 cm, with a perfect hemisphere of ice cream of the same radius (3 cm) sitting on top. Calculate the total external surface area of the dessert: the curved cone + the curved hemisphere only (no flat surfaces; the cone's base is sealed by the hemisphere). Give your answer in cm² to 1 d.p.

3 marks

Stuck? Cone slant l = √(3² + 10²) = √109. Cone CSA = πrl. Hemisphere curved = 2πr².

2. Explain your thinking

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

2.1 The lesson states that "a sphere has the smallest surface area for a given volume of any 3D shape — that is why nature uses spheres (bubbles, planets, cells)." Using this idea, explain (i) why industrial liquefied natural gas (LNG) storage tanks are often built as spheres rather than long cylinders, (ii) what trade-off engineers make if they choose a cylinder instead, and (iii) why a soap bubble is always spherical and not, say, cubic. Refer to surface area at least twice in your answer.

Stuck? Less surface area → less steel/material → less heat loss → less surface tension to overcome.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Hot-water cylinder (r = 0.27 m, h = 1.45 m)

(a) TSA = 2π(0.27)(1.45) + 2π(0.27)² = 0.783π + 0.1458π = 0.9288π ≈ 2.92 m².
(b) Cost = 2.92 × $32 = $93.44.

1.2 — Painting cylindrical tank (d = 1.4 m, h = 2.1 m)

(a) r = 0.7 m. TSA = 2π(0.7)(2.1) + 2π(0.7)² = 2.94π + 0.98π = 3.92π ≈ 12.32 m².
(b) Litres needed = 12.32 ÷ 8 = 1.54 ⇒ buy 2 litres (round up).

1.3 — Conical tent (d = 3 m, h = 2 m)

(a) r = 1.5 m. l = √(1.5² + 2²) = √(2.25 + 4) = √6.25 = 2.5 m.
(b) CSA = πrl = π(1.5)(2.5) = 3.75π ≈ 11.78 m².
(c) Cost = 11.78 × $28 ≈ $329.87 (exact: 105π ≈ $329.87).

1.4 — Spherical ornament wrap (d = 18 cm)

(a) r = 9 cm. SA = 4π(9)² = 324π ≈ 1017.88 cm².
(b) With 20% extra: 1017.88 × 1.20 ≈ 1221.46 cm².
(c) Cost = 1221.46 × $0.04 ≈ $48.86.

1.5 — Ice-cream cone + hemisphere (r = 3 cm, cone h = 10 cm)

Cone slant l = √(3² + 10²) = √109 ≈ 10.44 cm.
Cone CSA = πrl = π(3)(√109) ≈ 31.32π ≈ 98.39 cm².
Hemisphere curved SA = 2πr² = 2π(9) = 18π ≈ 56.55 cm².
Total external SA = 98.39 + 56.55 = 154.9 cm² (1 d.p.).

2.1 — Explain your thinking (sample response)

(i) LNG storage tanks are spheres because a sphere has the smallest surface area for a given volume. Less surface area means less expensive steel is needed and, more importantly, less heat can leak in or out through the walls — critical for keeping LNG cold and liquid.
(ii) If engineers choose a long cylinder instead (often for ease of transport or to fit on a truck), the trade-off is that the cylinder has a much larger surface area for the same volume. That means more steel, more weight, more cost, and more energy lost as the gas slowly warms.
(iii) A soap bubble is spherical because the soap film naturally pulls in to minimise its surface area. Surface tension acts like a stretched skin — the shape with the least skin per unit of air trapped inside is the sphere. A cube would have much more surface area for the same air volume, and so would have far more surface tension energy.

Marking: 1 for the "smallest SA for given V" principle, 1 for the cylinder trade-off (more material/cost or heat loss), 1 for the surface-tension reason for bubbles, 1 for clear sentence structure with "surface area" used at least twice.