Mathematics and statistics: TestAS practice testlets

When an object is scaled up or down while keeping its shape, its dimensions do not all change in the same proportion. If every length is multiplied by a scale factor k, every area, such as a surface or a cross-section, is multiplied by k², and every volume is multiplied by k³. Since mass is proportional to volume for a given material, mass also grows with k³. This relationship, known since Galileo, is called the square-cube law.

length × k area × k² volume and mass × k³ stress = force ÷ area ∝ k³ ÷ k² = k

The law has important consequences in engineering and biology. The load on a supporting cross-section, such as a leg, a column or a beam, grows with the weight, that is with k³, while the cross-section itself grows only with k². The stress in the material therefore grows in proportion to k. A statue that is scaled up by a factor of two, using the same material, experiences twice the stress in its legs. Scaled up far enough, any structure would collapse under its own weight, which is why giant animals must have disproportionately thick legs and why there is a limit to the height of buildings made of a given material.

The ratio of surface to volume decreases as objects grow: it is proportional to 1/k. Small objects have a large surface compared with their volume, large objects a small one. This explains why small animals lose heat quickly and must eat a lot relative to their mass, why fine powders burn or dissolve much faster than lumps, and why large engines and large buildings can be more energy-efficient than small ones.

Engineers use scale models in wind tunnels and towing tanks, but must take the square-cube law into account when transferring results to the full-size object. The paint needed for a ship's hull grows with the surface, the displacement of the ship with its volume. The questions refer to a statue and to a model of a ship built at a scale of 1:50, both made of the same materials as the originals. Effects such as the thickness of the paint layer are ignored.

Technical data are often given in units that are convenient in practice but not in the base units of the International System of Units (SI). Before calculating, all quantities should be converted into consistent units, usually metres, kilograms, seconds and the units derived from them, such as newtons, joules, watts and pascals. Errors in unit conversion are among the most frequent causes of wrong results, and in a few famous cases they have destroyed entire projects.

A reliable method is to multiply by fractions that equal one. Since 1 hour equals 3,600 seconds, the fraction 1 h ÷ 3,600 s equals one, and multiplying by it changes the unit without changing the quantity. For speeds, this gives the rule that a value in km/h is divided by 3.6 to obtain m/s. For energy, one kilowatt hour is 1,000 watts for 3,600 seconds, which equals 3,600,000 joules or 3.6 megajoules.

1 kWh = 3.6 MJ 1 bar = 100,000 Pa = 100 kPa 1 L = 1 dm³ = 0.001 m³ 1 m/s = 3.6 km/h 1 mm² = 10⁻⁶ m²

Areas and volumes need special care, because the conversion factor is squared or cubed. One metre is 1,000 millimetres, so one square metre is 1,000,000 square millimetres and one cubic metre is 1,000,000,000 cubic millimetres. One litre is one cubic decimetre, a thousandth of a cubic metre. Pressure is often given in bar, especially for tyres and hydraulic systems; one bar equals 100,000 pascals, roughly the air pressure at sea level. Density is often stated in grams per cubic centimetre; since one cubic metre contains a million cubic centimetres and a kilogram is a thousand grams, 1 g/cm³ equals 1,000 kg/m³.

Consumption figures can be expressed in two directions. Fuel consumption in Europe is usually given in litres per 100 kilometres, while in other countries the distance per unit of fuel is used, for example kilometres per litre or miles per gallon. The two are reciprocal: a high value of one means a low value of the other. After converting, a quick plausibility check helps: a result that suggests a car drives at 300 m/s in town or a kettle uses a megawatt hour should be recalculated. Writing the units next to every number in a calculation, and cancelling them like algebraic factors, is the simplest way to catch such mistakes early.