A-LevelMathematicsDifferentiation9709/12 · Oct/Nov 2019 · Q57 Marks

The diagram shows a solid cone which has a slant height of 15 cm and a vertical height of h cm. [Figure 5]

(i)Show that the volume, V cm³, of the cone is given by V = (1/3)π(225h – h³). [The volume of a cone of radius r and vertical height h is (1/3)πr²h.][2]
(ii)Given that h can vary, find the value of h for which V has a stationary value. Determine, showing all necessary working, the nature of this stationary value.[5]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.

[M1] Use of Pythagoras → r² = 15² - h²
[A1] V = 1/3π(225 - h²) × h → 1/3π(225h – h³) AG WWW e.g. sight of r = 15-h gets A0.
[B1] dV/dh = (π/3)(225 - 3h²)
[M1] Their dV/dh = 0 Differentiates, sets their differential to 0 and attempts to solve at least as far as h² ≠ 0.
[A1] (h =) √75, 5√3 or AWRT 8.66 Ignore -√75 OE and ISW for both A marks
[M1] d²V/dh² = (π/3)(-6h) = -2πh (-ve) Differentiates for a second time and considers the sign of the second differential or any other valid complete method.
[A1FT] → Maximum Correct conclusion from correct 2nd differential, value for h not required, or any other valid complete method. FT for their h, if used, as long as it is positive.
[SC] Omission of π or π/3 throughout can score B0M1A1M1A0

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