Describe — 5 marks
A student is designing a water tank for a school garden. The tank is cylindrical with a diameter measured as 1.2 m (to 1 d.p.) and a height measured as 2.8 m (to 1 d.p.). The student needs to estimate the volume and understand the range of possible values due to measurement uncertainty.
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(a) State the lower and upper bounds for the diameter of the tank.
[2 marks]
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(b) Describe how the bounds of the diameter and height affect the maximum possible volume of the tank. Use the formula V = πr²h in your explanation.
[2 marks]
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(c) The student estimates the volume as 3.17 m³ using the measured values. Describe why this estimate may not represent the actual volume that the tank can hold, referring to bounds in your answer.
[1 mark]
Show mark scheme
- (a) Lower bound for diameter: 1.15 m (1 mark)
- (a) Upper bound for diameter: 1.25 m (1 mark)
- (b) Maximum volume occurs when both radius and height are at their upper bounds (1 mark)
- (b) Using upper bound radius (0.625 m) and upper bound height (2.85 m) in V = πr²h gives the maximum possible volume / the upper bound of volume (1 mark)
- (c) The actual volume could be anywhere between the lower and upper bounds due to measurement uncertainty / the measured values are rounded so the true values could be different within the bounds (1 mark)
State — 2 marks
A student measures the length of a laboratory bench using a metre ruler. The ruler has divisions marked every 1 cm. The student records the length as 1.54 m.
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(a) State the smallest possible actual length of the bench.
[1 mark]
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(b) State the largest possible actual length of the bench.
[1 mark]
Show mark scheme
- (a) 1 mark for stating the lower bound as 1.535 m (or 153.5 cm)
- (b) 1 mark for stating the upper bound as 1.545 m (or 154.5 cm)
Explain — 4 marks
A physics teacher is planning a school trip to measure the height of a local building using a clinometer and measuring tape. The students need to estimate the uncertainty in their final result and understand how rounding affects their calculations.
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(a) A student measures the distance from the building as 24.7 m and the angle of elevation as 38.2°. When calculating the height using h = d × tan(θ), the student's calculator gives 19.319... m. Explain why the student should round this answer to 3 significant figures rather than using all the digits shown.
[2 marks]
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(b) The distance measurement has a lower bound of 24.65 m and an upper bound of 24.75 m. The angle has a lower bound of 38.15° and an upper bound of 38.25°. Calculate the upper bound for the height of the building.
[1 mark]
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(c) Explain why it would be inappropriate to report the final height of the building as 19.319 m given the precision of the measurements taken.
[1 mark]
Show mark scheme
- (a) The original measurements (distance and angle) were only measured to 3 significant figures / limited precision
- (a) The answer should not be given to more significant figures than the least precise measurement used in the calculation
- (b) h = 24.75 × tan(38.25°) = 24.75 × 0.7869... = 19.48 m (or 19.5 m to 3 s.f.) / correct calculation using upper bounds
- (c) The measurements have uncertainty/bounds, so reporting to 3 decimal places (0.001 m precision) falsely suggests greater accuracy than was actually achieved / the answer implies a precision greater than the input data allows
Explain — 2 marks
A student measures the length of a laboratory bench using a metre ruler. The ruler has markings at every centimetre. The student records the length as 1.5 m.
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(a) Explain why the recorded length of 1.5 m has a range of possible values.
[1 mark]
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(b) Write down the lower bound and upper bound for the length of the bench, given that it has been rounded to 1 decimal place.
[1 mark]
Show mark scheme
- (a) The measurement is rounded to 1 decimal place / the actual value could be anywhere within a range either side of 1.5 m / the ruler has a limit of precision (or similar valid explanation of measurement uncertainty)
- (b) Lower bound = 1.45 m AND upper bound = 1.55 m (both values must be correct for the mark)
Describe — 3 marks
A rectangular garden has length 8 metres and width 5 metres, measured to the nearest metre. A landscaper needs to calculate the area of the garden to order turf.
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(a) Describe how you would estimate the area of the garden using the rounded measurements.
[1 mark]
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(b) Describe how to find the lower bound and upper bound of the actual area of the garden.
[2 marks]
Show mark scheme
- (a) Multiply the rounded length by the rounded width (8 × 5 = 40 m²)
- (b) Lower bound found using 7.5 × 4.5 (= 33.75 m²)
- (b) Upper bound found using 8.5 × 5.5 (= 46.75 m²)
GCSE Perfect