Number 5
4 marks · HardA scientist measures the mass of a metal rod to be 756.7 g to the nearest gram and its length to be 120.5 cm to the nearest 0.1 cm.
Write down the lower and upper bounds for the mass and length measurements.
Calculate the minimum and maximum possible density of the rod in g/cm³. Density = Give both answers correct to 3 significant figures.
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(a) Mass: lower bound = 756.5 g, upper bound = 757.5 g; Length: lower bound = 120.45 cm, upper bound = 120.55 cm
(b) Minimum density = 756.5 ÷ 120.55 = 6.278 g/cm³ (to 3 s.f. = 6.28 g/cm³); Maximum density = 757.5 ÷ 120.45 = 6.288 g/cm³ (to 3 s.f. = 6.29 g/cm³)
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Marking points
M1 (a) · 1 mark · Correct identification of lower and upper bounds for mass and/or length.
A1 (a) · 1 mark · All four correct bounds clearly stated.
M2 (b) · 1 mark · Correct substitution into density formula using suitable bounds.
A2 (b) · 1 mark · Both minimum and maximum densities calculated correctly and rounded to 3 significant figures. (follow-through)
Worked solution
a) Lower bound for mass = 756.7 − 0.5 = 756.2 g, upper bound = 756.7 + 0.5 = 757.2 g. But since standard answers take 'nearest gram', bounds are 756.5 g (lower), 757.5 g (upper).
Lower bound for length = 120.5 − 0.05 = 120.45 cm, upper bound = 120.5 + 0.05 = 120.55 cm.
b) Minimum density = lowest mass ÷ highest length = 756.5 ÷ 120.55 = 6.278 g/cm³ (to 3 s.f. = 6.28 g/cm³).
Maximum density = highest mass ÷ lowest length = 757.5 ÷ 120.45 = 6.288 g/cm³ (to 3 s.f. = 6.29 g/cm³).