A-LevelMathematicsDifferentiation9709/12 · Feb/Mar 2016 · Q68 Marks

A vacuum flask (for keeping drinks hot) is modelled as a closed cylinder in which the internal radius is r cm and the internal height is h cm. The volume of the flask is 1000 cm³. A flask is most efficient when the total internal surface area, A cm², is a minimum.

(i)Show that A = 2πr² + 2000/r[3]
(ii)Given that r can vary, find the value of r, correct to 1 decimal place, for which A has a stationary value and verify that the flask is most efficient when r takes this value.[5]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2016). The mark or mark type is shown in brackets.

[B1] A = 2πr² + 2πrh
[M1] πr²h=1000 → h=1000/(πr²)
[A1] Substitute for h into A → A = 2πr² + 2000/r AG
[M1] Attempt differentiation & set = 0, e.g. dA/dr = 4πr - 2000/r² = 0
[A1] Correct differentiation and setting to 0
[DM1] Reasonable attempt to solve to r³, e.g. 4πr = 2000/r² → r³ = 2000/(4π) = 500/π
[A1] r = 5.4 (correct to 1 decimal place)
[B1] d²A/dr² = 4π + 4000/r³ > 0, hence MIN (Or convincing alternative method)

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