**Part (a)**

(Arc length $=$) $r\theta = r \times 1 = r$. Can be awarded by implication from later work, e.g. $3rh$ or $(2rh + rh)$ in the $S$ formula. (Requires use of $\theta = 1$). | B1 |

(Sector area $=$) $\frac{1}{2}r^2\theta = \frac{1}{2}r^2 \times 1 = \frac{r^2}{2}$. Can be awarded by implication from later work, e.g. the correct volume formula. (Requires use of $\theta = 1$). | B1 |

Surface area $= 2$ sectors $+ 2$ rectangles $+ $ curved face ($= r^2 + 3rh$) (See notes below for what is allowed here) | M1 |

Volume $= 300 = \frac{1}{3}r^2h$ | B1 |

Sub for $h$: $S = r^2 + 3 \times \frac{600}{r} = r^2 + \frac{1800}{r}$ | A1cso |

**Part (b)**

$\frac{dS}{dr} = 2r - \frac{1800}{r^2}$ or $2r - 1800r^{-2}$ or $2r + -1800r^{-2}$ | M1A1 |

$\frac{dS}{dr} = 0 \Rightarrow r^3 = ..., r = \sqrt[3]{900}$, or AWRT 9.7 (NOT $-9.7$ or $\pm 9.7$) | M1, A1 |

**Part (c)**

$\frac{d^2S}{dr^2} = ...$ and consider sign, $\frac{d^2S}{dr^2} = 2 + \frac{3600}{r^3} > 0$ so point is a minimum | M1, A1ft |

**Part (d)**

$S_{\min} = (9.65...)^2 + \frac{1800}{9.65...}$ (Using their value of $r$, however found, in the given $S$ formula) | M1 |

$= 279.65...$ (AWRT: 280) (Dependent on full marks in part (b)) | A1 |

**Guidance:**

**Part (a):**

M1 for attempting a formula (with terms added) for surface area. May be incomplete or wrong and may have extra term(s), but must have an $r^2$ (or $r^2\theta$) term and an $rh$ (or $rh\theta$) term.

**Part (b):**

In parts (b), (c) and (d), ignore labelling of parts

$1^{\text{st}}$ M1 for attempt at differentiation (one term is sufficient) $r^n \to kr^{n-1}$

$2^{\text{nd}}$ M1 for setting their derivative (a 'changed function') $= 0$ and solving as far as $r^3 = ...$ (depending upon their 'changed function', this could be $r = ...$ or $r^2 = ...$, etc., but the algebra must deal with a negative power of $r$ and should be sound apart from possible sign errors, so that $r^n = ...$ is consistent with their derivative).

**Part (c):**

M1 for attempting second derivative (one term is sufficient) and considering its sign. Substitution of a value of $r$ is not required. (Equating it to zero is M0).

A1ft for a correct second derivative (or correct ft from their first derivative) and a valid reason (e.g. $> 0$), and conclusion. The actual value of the second derivative, if found, can be ignored. To score this mark as ft, their second derivative must indicate a minimum.

**Alternative:**

M1: Find value of $\frac{dS}{dr}$ on each side of their value of $r$ and consider sign.

A1ft: Indicate sign change of negative to positive for $\frac{dS}{dr}$, and conclude minimum.

**Alternative:**

M1: Find value of $S$ on each side of their value of $r$ and compare with their 279.65.

A1ft: Indicate that both values are more than 279.65, and conclude minimum.

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