Higher June 2022 Paper 4 Q7
7 The diagram shows a square-based pyramid and a sphere.
The pyramid has base length 12.3 cm and perpendicular height 15.7 cm.
The sphere has radius \(r\) cm.
The pyramid and the sphere have the same volume.
Work out the radius of the sphere.
You must show your working.
[The volume of a pyramid is \(\frac{1}{3}\) × area of base × perpendicular height.
The volume \(V\) of a sphere with radius \(r\) is \(V = \frac{4}{3}\pi r^3\).] [5]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 5.73 to 5.74 or 5.7 with correct working | 5 | “correct working” requires at least M3 or if trials are used M1 M1 M1 | |
| M4 for [\(r^3 =\)] \(\frac{1}{3} \times 12.3^2 \times 15.7 \div \frac{4}{3}\pi\) oe or for [\(r^3 =\)] their \(791.8 \div \frac{4}{3}\pi\) OR M3 for \(\frac{4}{3}\pi r^3 = \left(\frac{1}{3} \times 12.3^2 \times 15.7\right)\) oe or \(\frac{4}{3}\pi r^3 =\) their 791.8 OR M1 for \(\frac{1}{3} \times 12.3^2 \times 15.7\) oe A1 for 791.7 to 791.8 or 792 Trials We need value and its result for M1 M1 for \(\frac{1}{3} \times 12.3^2 \times 15.7\) oe A1 for 791.7 to 791.8 or 792 M1 for a correct trial M1 for another correct trial | Notes: allow 151[.29] or 151.2 or 151.3 for \(12.3^2\), for \(\frac{1}{3}\) accept 0.33 or better and for \(\frac{4}{3}\) accept 1.33 or better and for \(\frac{4}{3}\pi\) accept 4.17 to 4.19 their 791.8 is 791.7 to 791.8 or 792 or from correct use of given formula M3 implied by \(V = \frac{4}{3}\pi r^3\) and \(V =\) their 791.8 do not lose M1 A1 if further work on a sphere does not include this | ||
| If 0, 1 or 2 scored instead award SC3 for answer 5.73 to 5.74 or 5.7 with no working or insufficient working If 0 or 1 scored instead award SC2 for 188.99 to 189.03 with no working or insufficient working If 0 scored SC1 for 791.7 to 791.8 or 792 with no working or insufficient working | |||