Higher June 2025 Paper 3 Q21
21 Five identical spheres just fit inside a cylinder.
Each sphere has radius \(r\).
| Volume of a sphere \(= \dfrac{4}{3}\pi r^3\) |
What fraction of the space inside the cylinder is filled by the spheres?
You must show your working. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1: uses \(r\) as a variable | ||
| \(5 \times 2r\) or \(10r\) | M1 | may be seen on diagram or as part of a calculation |
| \((\pi)r^2 \times\) their \(10r\) or \(10(\pi)r^3\) | M1 | oe their \(10r\) must be of the form \(kr\) |
| \(\dfrac{5 \times \dfrac{4}{3}(\pi)r^3}{(\pi)r^2 \times \text{their } 10r}\) | M1dep | oe eg \(\dfrac{\;\dfrac{20}{3}\;}{10}\) or \(\dfrac{20}{3} \div 10\) must be seen consistently with or without \(\pi\) to match cylinder formula dep on 2nd M1 |
| \(\dfrac{2}{3}\) with M3 awarded | A1 | oe fraction eg \(\dfrac{20}{30}\) |
| Alternative method 2: uses a value for \(r\) | ||
| Method to calculate the cylinder length using their value of \(r\) | M1 | their value of \(r\) must be stated or may be implied on the diagram or from their working eg \(r = 1\) and cylinder length = 10 |
| Substitutes the same value of \(r\) and their cylinder length into \((\pi)r^2h\) | M1dep | the substitution must be shown accept without use of \(\pi\) eg \(\pi \times 1^2 \times 10\) or \(1^2 \times 10\) |
| Substitutes the same value of \(r\) into \(5 \times \dfrac{4}{3}(\pi)r^3\) and writes a fraction with their volume for the 5 spheres over their volume for the cylinder | M1dep | the substitution must be shown must be seen consistently with or without \(\pi\) to match cylinder formula eg \(5 \times \dfrac{4}{3} \times \pi \times 1^3\) and \(\dfrac{\;\dfrac{20\pi}{3}\;}{10\pi}\) dep on M2 |
| \(\dfrac{2}{3}\) with M3 awarded | A1 | oe fraction eg \(\dfrac{20}{30}\) |
Additional guidance
| (Length =) \(5r\) \(\pi r^2 \times 5r\) \(\dfrac{5 \times \dfrac{4}{3}\pi r^3}{\pi r^2 \times 5r}\) | M0M1M1 |
| Condone \(\dfrac{66.\dot{6}}{100}\) for \(\dfrac{2}{3}\) | |
| Do not accept a misread for the volume of a sphere | |
| Ignore any attempt to convert the correct fraction to a decimal or percentage | |
| Ignore any attempt to simplify the fraction once the correct answer has been seen |