AQA GCSE Mathematics Paper 3 (Higher), June 2025: Question 21

4 marks · Medium difficulty · Multi-step Problem

Find the fraction of the space inside a cylinder that is filled by five identical spheres of radius r that just fit inside it.

Practise this question

Question

Question 21 asks to find what fraction of the space inside a cylinder is filled by five identical spheres that just fit inside it end-to-end. A diagram illustrates the five spheres tightly packed inside the horizontal cylinder. A formula box provides: Volume of a sphere = 4/3 pi r^3. Working space and an answer line worth 4 marks are provided.

Mark scheme

Show the mark scheme Mark scheme for Question 21 outlining two alternative methods. Alternative 1 uses variable r: M1 for cylinder length 10r, M1 for cylinder volume pi * r^2 * 10r, M1dep for fraction [5 * 4/3 pi r^3] / [pi * r^2 * 10r], and A1 for the simplified answer 2/3. Alternative 2 uses a numerical value for r (e.g., r = 1) with equivalent method steps to achieve 2/3. Additional guidance and notes are given.

How to answer it

Volume Ratio: Spheres Inside a Cylinder

What this question tests

This question assesses your ability to deduce geometric dimensions from a diagram, apply 3D volume formulas (cylinder and sphere), link diameter to radius, set up algebraic ratios, and simplify complex fractions involving π and powers of variables.

Question 21 • 4 Marks

Full Question Analysis & Step-by-Step Solution

Five identical spheres of radius r just fit inside a cylinder. What fraction of the cylinder's volume is filled by the spheres?

💡 Key Knowledge Required

  • Volume of a sphere: V = (4/3)πr³ (given on paper).
  • Volume of a cylinder: V = πr²h (must recall).
  • "Just fits" meaning:
    • Cylinder radius = sphere radius = r
    • Sphere diameter = 2r
    • Total length/height = 5 × diameter = 5 × 2r = 10r

🧠 Top Exam Techniques

  • Two valid approaches: You can work purely algebraically with r , or pick a simple value (e.g. let r = 1 ). Both get full marks!
  • Leave π in terms of π: Do not evaluate 3.14159... on your calculator. It will cancel out completely in the fraction.
  • Show all substitutions: The mark scheme requires explicitly visible formulas to unlock method marks.

📐 Method 1: Algebraic Solution (Step-by-Step)

Step 1: Determine the length (height) of the cylinder
Each sphere touches the next and fits end-to-end. One sphere has a diameter of 2r .
Length of cylinder, h = 5 × 2r = 10r
Mark 1 (M1): Awarded for finding 5 × 2r or 10r (seen in working or clearly marked on the diagram).
Step 2: Calculate the total volume of the cylinder
Cross-sectional radius = r , height = 10r .
Volume of cylinder = π × r² × h = π × r² × 10r = 10πr³
Mark 2 (M1): Awarded for substituting their length into the cylinder volume formula: π × r² × (10r) = 10πr³ .
Step 3: Calculate the total volume of all 5 spheres
Volume of 1 sphere = (4/3)πr³
Volume of 5 spheres = 5 × (4/3)πr³ = (20/3)πr³
Step 4: Form the fraction and simplify
Fraction = (Volume of 5 spheres) ÷ (Volume of cylinder)
Fraction = [(20/3)πr³] / [10πr³]
Notice that π and r³ cancel from both numerator and denominator:
Fraction = (20/3) ÷ 10 = 20 / 30 = 2/3
Mark 3 (M1dep): Awarded for setting up the division fraction: [5 × (4/3)πr³] / [10πr³] .
Mark 4 (A1): Awarded for simplifying to 2/3 (or equivalent fraction, e.g. 20/30 ).

📐 Method 2: Smart Substitution (Let r = 1)

Because the question asks for a fraction, the actual scale does not matter. You can choose any convenient number for radius r :

  • Let r = 1 cm.
  • Sphere diameter = 2 × 1 = 2 cm → Cylinder length = 5 × 2 = 10 cm.
  • Volume of cylinder = π × 1² × 10 = 10π cm³.
  • Volume of 5 spheres = 5 × (4/3) × π × 1³ = (20/3)π cm³.
  • Fraction = [(20/3)π] / [10π] = (20/3) / 10 = 2/3 .

✅ Final Answer

2/3

Acceptable equivalents: 20/30 or recurring decimal equivalent 0.666... or 66.6̇% . Leaving it as a simplified fraction is cleanest and least error-prone.

❌ Common Mistakes to Avoid

  • Using 5r instead of 10r for length: Multiplying radius r by 5 gives only half the cylinder length! Each sphere takes up 2r of length.
  • Forgetting to multiply sphere volume by 5: There are 5 spheres inside, not just one.
  • Formula confusion: Using cylinder surface area or circumference formula instead of volume ( πr²h ).
  • Premature rounding: Converting intermediate fractions to rounded decimals (e.g. 0.67), which leads to an inaccurate final ratio.

Topics

Geometry and measures · Ratio, proportion and rates of change · 3.4.2 Mensuration and calculation · 3.3 Ratio, proportion and rates of change

Question and mark scheme from the AQA GCSE Mathematics examination, Paper 3 (Higher), June 2025. QuestionVault is an independent revision resource; questions remain the copyright of the awarding body.