Show — 4 marks
A swimming pool maintenance company needs to calculate the volume of water in a rectangular pool to determine how much chlorine treatment is needed. The pool has a length of 25 m, a width of 10 m, and a uniform depth of 2 m.
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(a) Show that the volume of water in the pool is 500 m³.
[2 marks]
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(b) The pool is drained and refilled. During refilling, the water level rises at a rate of 0.5 m per hour. Show that it takes 4 hours to fill the pool to a depth of 2 m.
[1 mark]
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(c) A rectangular paddling pool next to the main pool has dimensions 8 m × 6 m × 0.8 m. Calculate the total volume of water needed to fill both pools completely. Show your working.
[1 mark]
Show mark scheme
- (a) Correct formula used: V = length × width × depth (1 mark)
- (a) Correct substitution and calculation: 25 × 10 × 2 = 500 m³ (1 mark)
- (b) Correct calculation: 2 m ÷ 0.5 m/hour = 4 hours (1 mark)
- (c) Correct total volume: paddling pool volume (8 × 6 × 0.8 = 38.4 m³) added to main pool volume (500 m³) = 538.4 m³ (1 mark)
State — 5 marks
A physics laboratory is designing a cylindrical water tank to store distilled water for experiments. The tank has an internal diameter of 0.80 m and a height of 1.5 m. The laboratory manager needs to understand the tank's capacity and surface area for ordering materials and calculating water storage.
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(a) State the formula for the volume of a cylinder.
[1 mark]
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(b) Calculate the volume of the water tank in m³. State your answer to 2 significant figures.
[2 marks]
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(c) The curved surface area of the cylinder is given by the formula A = 2πrh. State the curved surface area of the tank in m². State your answer to 2 significant figures.
[2 marks]
Show mark scheme
- (a) V = πr²h (or equivalent notation with π × r² × h or π × r × r × h)
- (b) Correct substitution: π × 0.4² × 1.5 (or π × 0.16 × 1.5) [1 mark]
- (b) Final answer: 0.75 m³ or 0.76 m³ to 2 s.f. [1 mark]
- (c) Correct substitution: 2 × π × 0.4 × 1.5 (or 2π × 0.4 × 1.5) [1 mark]
- (c) Final answer: 3.8 m² to 2 s.f. [1 mark]
Define — 2 marks
A civil engineer is designing a cylindrical water storage tank for a small village. The tank will have a diameter of 4 metres and a height of 6 metres. Before construction begins, the engineer needs to ensure the team understands the key measurements involved.
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(a) Define what is meant by the term 'volume' in the context of the water storage tank.
[1 mark]
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(b) Define what is meant by the term 'cross-sectional area' of the cylindrical tank.
[1 mark]
Show mark scheme
- (a) Volume is the amount of space occupied by / contained within the tank (or the three-dimensional space it takes up)
- (b) Cross-sectional area is the area of the circular face / the area you would see if you cut through the cylinder perpendicular to its height
Describe — 3 marks
A manufacturer produces two cuboid storage containers. Container P has dimensions 10 cm × 6 cm × 5 cm. Container Q is mathematically similar to Container P, but each dimension is twice as long.
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(a) Describe how to calculate the volume of Container P.
[1 mark]
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(b) Describe how the volume of Container Q compares to the volume of Container P.
[2 marks]
Show mark scheme
- (a) Multiply the three dimensions together (10 × 6 × 5) or reference to length × width × height
- (b) State that the volume of Q is 8 times the volume of P (or equivalent)
- (b) Explain that doubling all three dimensions means 2 × 2 × 2 = 8 (or 2³)
Calculate — 5 marks
A rectangular water tank has a square base with side length 60 cm. The tank is initially empty and has a height of 90 cm.
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(a) Calculate the volume of the tank in cubic metres. Give your answer in standard form.
[2 marks]
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(b) Water is poured into the tank to a depth of 45 cm. Calculate the total surface area of the tank that is in contact with the water.
[3 marks]
Show mark scheme
- (a) Volume = 0.6 × 0.6 × 0.9 or 60 × 60 × 90 seen
- (a) 3.24 × 10⁻¹ m³ or equivalent
- (b) Area of base = 0.6 × 0.6 = 0.36 m²
- (b) Area of four sides = 4 × (0.6 × 0.45) = 1.08 m²
- (b) Total surface area = 1.44 m²
GCSE Perfect