Practice Maths

L21 — Applications of Measurement

Key Terms

Area unit conversion
Square the length conversion factor: 1 m = 100 cm, so 1 m² = 100² = 10 000 cm².
Volume unit conversion
Cube the length conversion factor: 1 m = 100 cm, so 1 m³ = 100³ = 1 000 000 cm³.
Composite shape
Decompose into simpler solids, calculate each part, then add (joined) or subtract (hollowed-out).
Scaling
If dimensions scale by k: length × k, area × k², volume × k³.
Density
Mass per unit volume. Mass = density × volume. Units: kg/m³ or g/cm³.
Flow rate
Volume per unit time. Volume = flow rate × time. Units: m³/s, L/min, etc.

Unit Conversions

AreaVolume / Capacity
1 cm² = 100 mm² 1 cm³ = 1000 mm³
1 m² = 10 000 cm² 1 m³ = 1 000 000 cm³
1 ha = 10 000 m² 1 L = 1000 cm³ = 1000 mL
1 km² = 1 000 000 m² 1 m³ = 1000 L

Key insight: if a length unit is scaled by k, area scales by k² and volume by k³.

Composite Shapes Strategy

  1. Decompose the shape into simpler solids.
  2. Calculate each part separately.
  3. Add parts (if joined) or subtract parts (if hollowed out).
  4. Be careful not to count shared faces twice in surface area.

Scaling

Scale factor kLengthAreaVolume
2×2×4×8
3×3×9×27
k×k×k²×k³
Hot Tip: When scaling: length is ×k, area is ×k², volume is ×k³. Apply the power that matches the number of dimensions. A common error is using k for area or k² for volume — always check what kind of measurement you are scaling.

Unit Conversions

When converting units for area or volume, apply the length-unit conversion twice (for area) or three times (for volume).

Worked Example 1 — Unit conversion

Convert 2.5 m² to cm².

1 m = 100 cm, so 1 m² = 100² cm² = 10 000 cm².

2.5 m² = 2.5 × 10 000 = 25 000 cm².

Composite Shapes

Worked Example 2 — Swimming pool (trapezoidal prism)

A pool is 25 m long and 8 m wide. The depth varies: 1 m at the shallow end, 3 m at the deep end.

Cross-section is a trapezium: parallel sides 1 m and 3 m, length 25 m.

Across-section = ½(1+3)(25) = 50 m². V = 50 × 8 = 400 m³ = 400 000 L.

Scaling

Worked Example 3 — Scaling volumes

A model car is built at scale 1:20. If the real car has a fuel tank of 60 L, what is the volume of the model tank?

Length scale: 1:20. Volume scale: 1:20³ = 1:8000.

Model volume = 60 000 mL / 8000 = 7.5 mL.

  1. Unit conversions. Fluency

    • (a) Convert 3.5 m² to cm².
    • (b) Convert 85 000 cm² to m².
    • (c) Convert 0.6 m³ to litres.
    • (d) A lake has area 4.2 km². Convert to hectares. (1 km² = 100 ha)

  2. Composite surface areas. Fluency

    • (a) An L-shaped cross-section wall: the overall shape is 8 m wide and 5 m tall; a 5 m wide × 3 m tall rectangular notch is cut from the top right. Find the area of the L-shaped cross-section.
    • (b) A circle of radius 4 cm has a square hole of side 3 cm punched through its centre. Find the remaining area.
    • (c) A rectangular room (6 × 4 m) has walls 2.8 m high. There are two windows (1.2 × 1.0 m each) and one door (0.9 × 2.1 m). Find the wall area to be painted.
    • (d) An annulus (ring): outer radius 10 cm, inner radius 6 cm. Find the area of the ring.

  3. Composite volumes. Fluency

    • (a) A solid cylinder (r=5 cm, h=12 cm) has a conical hole (r=5 cm, h=12 cm) drilled into it from the top. Find the remaining volume.
    • (b) A hemisphere (r=6 cm) is placed on top of a cone (r=6 cm, h=8 cm) of the same base. Find the total volume.
    • (c) An L-shaped prism: the cross-section is an L-shape made from a 10×8 cm rectangle with a 6×5 cm rectangle removed. The prism is 4 cm deep. Find the volume.
    • (d) A rectangular tank (50×30×40 cm) is half-filled with water. A solid sphere of radius 6 cm is dropped in. By how much does the water level rise? (Give answer to 2 d.p.)

  4. Real-world contexts. Fluency

    • (a) Grass seed covers 40 g per m². A rectangular lawn is 12 m × 8 m. How many kilograms of seed are needed?
    • (b) Tiles are 30 cm × 30 cm. A bathroom floor is 3.6 m × 2.4 m. How many tiles are needed? (Allow 10% extra for cuts.)
    • (c) A cubic metre of topsoil costs $65. How much does it cost to cover a garden bed (4 m × 3 m) to a depth of 15 cm?
    • (d) A rainwater tank is a cylinder (r=0.8 m, h=1.5 m). Rain falls at a rate of 12 mm/h. The collection area (roof) is 80 m². How long (to the nearest hour) to fill the tank completely?

  5. Swimming pool cross-section. Understanding

    The diagram shows a cross-section of a swimming pool (side view). The pool is 25 m long, 10 m wide, and has a sloping floor.

    1.2 m 3.0 m 25 m width = 10 m (into page)
    • (a) What shape is the cross-section of the pool? Name it and identify the parallel sides.
    • (b) Find the area of the cross-section.
    • (c) Find the volume of water in the pool in m³.
    • (d) Express the volume in litres. If a pump removes water at 500 L/min, how long (in hours and minutes) to empty the pool?

  6. Scaling. Understanding

    • (a) All dimensions of a rectangular prism are doubled. By what factor does the volume increase?
    • (b) A sphere has its radius tripled. What is the ratio of the new surface area to the original?
    • (c) A model building is built at scale 1:50. The real building’s roof area is 600 m². Find the model roof area in cm².
    • (d) Two cylinders are similar. The larger has volume 500 cm³ and radius 5 cm; the smaller has radius 2 cm. Find its volume.

  7. Cost and material problems. Understanding

    • (a) A wall is 8 m long, 2.5 m high. Render costs $15/m². There is one window (1.5×1 m). Find the cost to render.
    • (b) A cylindrical water tank (r=1 m, h=2 m) needs to be painted on the outside (curved surface + one base). Paint covers 12 m²/L. How many litres are needed to 1 d.p.?
    • (c) A concrete path surrounds a rectangular swimming pool (20×8 m). The path is 1.5 m wide on all sides and 10 cm deep. Find the volume of concrete needed.
    • (d) Gold is shaped into a sphere of radius 1.5 cm. It is then reshaped into a wire of radius 0.5 mm (0.05 cm). Find the length of wire produced.

  8. Rates and flow. Understanding

    • (a) Water flows through a pipe (circular cross-section, radius 2 cm) at 0.5 m/s. Find the volume of water per second (in cm³/s).
    • (b) A conical tank (apex at the bottom, r=3 m at the top, h=4 m) is being filled at 0.5 m³/min. How long to fill it?
    • (c) A sphere of ice (r=10 cm) melts at a rate that reduces the radius by 0.5 cm per hour. What is the surface area when the radius has halved?
    • (d) Sand falls from a conveyor into a conical pile. The pile always has r = h. When h=60 cm, find the volume and surface area of the pile. (Use l = h√2 for slant height when r=h.)

  9. Grain silo. Problem Solving

    A grain silo consists of a cylinder (r=3 m, h=8 m) with a hemispherical dome on top.

    • (a) Find the total volume of the silo.
    • (b) Find the total outer surface area of the silo. (The bottom of the cylinder is on the ground — include it.)
    • (c) The silo is filled to 90% capacity. How many tonnes of grain does it hold if wheat has density 780 kg/m³?
    • (d) A second silo has the same total height (8 + 3 = 11 m) but no dome (pure cylinder). If it has the same volume as the original silo, find its radius to 2 d.p.

  10. Olympic athletics track. Problem Solving

    An athletics track consists of two straight sections (each 84.39 m long) and two semicircular ends. The inner edge of lane 1 has radius 36.8 m for the semicircles. Each lane is 1.22 m wide.

    • (a) Find the total length of the inside edge of lane 1 (the standard 400 m track). (Verify: 2×84.39 + 2π×36.8 ≈ 400 m)
    • (b) Find the total area enclosed by the inner edge of lane 1.
    • (c) Find the total area of all 8 lanes combined (to the nearest m²).
    • (d) The track surface is made of 12 mm of rubber compound. Find the volume of rubber required for all 8 lanes in m³.
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