Higher June 2024 Paper 5 Q18
18 A sphere has radius \(x\) cm.
A cone has radius \(R\) cm and height \(2R\) cm.
The volume of the sphere is equal to the volume of the cone.
Write \(R\) in terms of \(x\).
[The volume \(V\) of a sphere with radius \(r\) is \(V = \frac{4}{3}\pi r^3\).
The volume \(V\) of a cone with radius \(r\) and height \(h\) is \(V = \frac{1}{3}\pi r^2 h\).] [4]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| \(R = \sqrt[3]{2}\,x\) or \(R = \sqrt[3]{2x^3}\) final answer | 4 | Condone \(r\) for \(R\) throughout Answer \(\sqrt[3]{2}\,x\) or \(\sqrt[3]{2x^3}\) allow M3 For method marks condone 3.14, 3.142 or \(\frac{22}{7}\) for \(\pi\) For method marks ignore any units | |
| M3 for \(12[\pi]x^3 = 6[\pi]R^3\) or better Or M2 for \(\frac{4}{3}[\pi]x^3 = \frac{1}{3}[\pi] \times R^2 \times 2R\) oe | For M3 removes fractions and simplifies terms in \(R\) M3 accept e.g. \(2x^3 = R^3\) | ||
| or M1 for \(\frac{1}{3}\pi \times R^2 \times 2R\) | Must see correct expression in \(R\) for M1 | ||