Live practice

Zinc is more reactive than tin and could react more readily with acids in food, forming salts that contaminate the contents. Tin is less reactive and provides a safer protective barrier between the food and the iron can.

Plaster of Paris is calcium sulphate hemihydrate. It takes up water and changes into hard gypsum:

Equivalently, integer coefficients may be used by doubling every formula unit.

Principle. Newton cooling makes the excess temperature θ=T−Ts fall exponentially: θ=θ0e^(−kt).

Principle. A soap bubble has two interfaces, so ΔP=4S/r. An air bubble in liquid has one interface, so its absolute pressure is Patm+ρgh+2S/r.

Principle. Conserve the two-star system's mechanical energy. At contact the centre separation is twice the stellar radius and both stars have equal speed.

Principle. Impulse is the vector change in momentum. Here the ball turns through 45 degrees from its incident line, so the angle between the incoming and outgoing velocity vectors is 135 degrees.

Principle. Definitions using endpoint differences are always true; formulas obtained by integrating a constant acceleration require that extra condition.

Define the sample space and count each favourable outcome once.

The final value lies in [0,1]; complement pairs also sum to one, providing an independent check.

Form the required totals explicitly and divide by the full number of observations.

The variance is nonnegative and the reported standard deviation is its positive square root.

Simplify the expression before substitution and preserve one-sided information.

Substitution into the original expression verifies the stated domain and final value.