Number 5

4 marks · Hard

The diameter of a pipe is measured as 8.60 cm to the nearest 0.05 cm. The length of the pipe is given as 2.00 m to the nearest 1 cm.

The volume of the pipe is calculated using the formula: \[ V = \pi r^2 h \] where is the radius (in cm) and is the length (in cm). Calculate the lower and upper bounds for the volume of the pipe, giving your answers to 3 significant figures.

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Lower bound: use smallest radius and shortest height; upper bound: use largest radius and greatest height. Calculations yield lower bound ≈ 1150 cm³, upper bound ≈ 1170 cm³ (both to 3 s.f.).

Also accept: Alternative method for bounds calculation – awarding marks for correct method.

Common mistakes: Forgetting to divide diameter by 2 for radius.; Incorrect application of upper/lower bound rules for products.

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Marking points

M1 · 1 mark · Correctly finds upper/lower bounds of diameter and length.

M2 · 1 mark · Correctly finds upper/lower bounds for radius (halves diameter bounds).

M3 · 1 mark · Correct use of V = for either bound.

A1 · 1 mark · Both bounds calculated correctly to 3 significant figures, with working for at least one full calculation. (follow-through)

Worked solution

Diameter lower bound = 8.60 – 0.025 = 8.575 cm; diameter upper bound = 8.60 + 0.025 = 8.625 cm.

Radius lower bound = 8.575 ÷ 2 = 4.2875 cm; radius upper bound = 8.625 ÷ 2 = 4.3125 cm.

Length lower bound = 200 cm – 0.5 = 199.5 cm; length upper bound = 200 cm + 0.5 = 200.5 cm.

Lower bound for volume = cm³ (rounded to 3 s.f.: 1150 cm³).

Upper bound for volume = cm³ (rounded to 3 s.f.: 1170 cm³).