Higher November 2022 Paper 4 Q12
12 The diagram shows a sphere and a cone.
Not to scale
The sphere has radius 4 cm.
The cone has radius 15 cm and height 30 cm.
The sphere is completely filled with water.
The same amount of water is poured into the cone.
Work out the depth, \(d\) cm, of the water in the cone.
You must show your working.
[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 \(V = \frac{1}{3}\pi r^2 h\).] [6]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 10.1 or 10.07 to 10.08 with correct working | 6 | M4 for \(\frac{4}{3}\) × π × \(4^3\) = \(\frac{1}{3}\) × π × \((\frac{1}{2}d)^2\) × \(d\) oe M1 for rearranging their equation OR M1 for \(\frac{4}{3}\) × π × \(4^3\) implied by 268.[08…] or 268.1 B1 for [radius of water =] \(\frac{1}{2}d\) M2 for \(\frac{1}{3}\) × π × \((\frac{1}{2}d)^2\) × \(d\) = their 268.08… or M1 for \(\frac{1}{3}\) × π × \((\frac{1}{2}d)^2\) × \(d\) condone one error M1 for rearranging their equation e.g. \(d^3 = \frac{their\ 268.08\ldots \times 12}{\pi}\) implied by [\(d^3\) =] 1024 Alternative method M4 for \(\frac{\frac{256\pi}{3}}{\frac{1}{3}\pi \times 15^2 \times 30} = \left(\frac{d}{30}\right)^3\) oe M1 for rearranging their equation OR M1 for \(\frac{4}{3}\) × π × \(4^3\) implied by 268.[08…], 268.1 or \(\frac{256\pi}{3}\) M1 for \(\frac{1}{3}\)π × \(15^2\) × 30 implied by 2250π or 7068[.58…] or 7069 or 7068.6 M2 for [sf=] \(\sqrt[3]{\frac{\frac{256\pi}{3}}{\frac{1}{3}\pi \times 15^2 \times 30}}\) oe or 0.33597… rot or M1 for \(\frac{\frac{256\pi}{3}}{\frac{1}{3}\pi \times 15^2 \times 30}\) oe or 0.0379… M1 for their sf × 30 If 0 or 1 scored, instead award SC3 for answer 10.1 or 10.07 to 10.08 with no or insufficient working | allow 10 only if working supports it “Correct working” requires evidence of at least M2 or M1 M1 M1 implied by \(\frac{256\pi}{3}\) Note they can use \(h = 2r\) so apply the scheme accordingly, 1024 will become 256 Complete method with trials giving the correct answer can score 6 marks M1 for each correct trial up to a maximum of M3 |