**(a)**

| | Shell | Wax | Filled shell |
|---|---|---|---|
| Mass ratio | $m$ | $3m$ | $4m$ |
| Dist. above vertex | $\frac{2}{3} \times 6r$ | $\frac{3}{4} \times 2r$ | $\overline{x}$ |

| B1 |

$4mr + \frac{9}{2}mr = 4m\overline{x}$ | M1A1ft |

$\overline{x} = \frac{17}{8}r$ | A1 | (4)

**Notes for Question 3(a):**

B1 | for correct distances from the vertex or any other point |

M1 | for a dimensionally correct moments equation with their distances and masses |

A1 ft | for a correct moments equation, follow through their distances but must have correct masses |

A1 cso | for $\overline{x} = \frac{17}{8}r$ |

NB: If $\frac{2}{3}$ and $\frac{3}{4}$ are interchanged in the distances, the correct answer is obtained but the solution is incorrect. Score: B0M1A1A0 |

**(b)**

$\tan \theta = \frac{r}{6r - \overline{x}} = \frac{r}{31r/8}$ | M1A1ft |

$\tan \theta = \frac{8}{31}$ |  |

$\theta = 14.47...°$ | A1 | (3) [7]

**Notes for Question 3(b):**

M1 | for $\tan \theta = \frac{r}{6r - \overline{x}}$. Can be either way up, but must include $6r - \overline{x}$. Substitution for $\overline{x}$ not required |

A1 ft | for $\tan \theta = \frac{r}{31r/8}$ oe ft their $\overline{x}$ |

for $\theta = 14.47...°$ Accept 14°, 14.5° or better or $\theta = 0.2525...$ rad |

A1 cso | Accept 0.25 or better. Obtuse angle accepted. |

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