# Question 3:

## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\pi r^2 h=16000\pi$ | B1 | Correct equation for volume seen or used; $h$ likely to be used, but could be other variable |
| $A=2\pi r^2+2\pi rh$ | B1 | Correct expression for surface area seen; two terms may be seen at separate stages; if alternative formula used eg $2\pi r^2+2Vr^{-1}$ then must be clearly derived; allow BOD for $2\pi r^2+2\pi r\times16000r^{-2}$ as long as $h$ seen explicitly in terms of $r$ first |
| $=2\pi r^2+2\pi r\times16000r^{-2}$ | M1 | Eliminate $h$ from expression for surface area; allow if just attempt at curved surface area |
| $=2\pi r^2+32000\pi r^{-1}$ **A.G.** | A1 | Obtain given answer; if $2\pi r^2$ is first seen in the final answer then it must be justified eg 'plus two ends', otherwise max B1B0M1A0 |
| **[4]** | | |

## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{dA}{dr}=4\pi r-32000\pi r^{-2}$ | M1 | Attempt differentiation; both powers decrease by 1 |
| $4\pi r-32000\pi r^{-2}=0$ | M1 | Equate derivative to 0 and attempt to solve for $r$ (or $h$); $-32000\pi h^{-2}+\pi\sqrt{16000}h^{-\frac{1}{2}}$ |
| $r^3=8000$ | | $h^{\frac{3}{2}}=\sqrt{64000}$ |
| $r=20$ | A1 | Obtain correct $r$, units not needed |
| Surface area $=2400\pi$ cm$^2$ / 7540 cm$^2$ | A1 | Obtain correct $A$, units not needed; allow exact or decimal (3sf or better) |
| $\frac{d^2A}{dr^2}=4\pi+64000\pi r^{-3}$ | M1 | Attempt method to justify minimum, including substitution or consideration of sign; could also test first derivative, or $A$, on both sides of $r=20$ |
| when $r=20$, $\frac{d^2A}{dr^2}=12\pi$ (or 37.7) | | |
| $\frac{d^2A}{dr^2}>0$, hence minimum | A1 | Correct conclusion, with justification, from correct working; if second derivative is evaluated, it must be correct (condone truncated decimal of 37.6) |
| **[6]** | | |

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