A-LevelMathematicsDifferentiation9709/12 · Feb/Mar 2017 · Q35 Marks

The diagram shows a water container in the form of an inverted pyramid, which is such that when the height of the water level is h cm the surface of the water is a square of side ½h cm. [Figure 3.1] [The volume of a pyramid having a base area A and vertical height h is Ah.]

(i)Express the volume of water in the container in terms of h.[1]
(ii)Water is steadily dripping into the container at a constant rate of 20 cm³ per minute. Find the rate, in cm per minute, at which the water level is rising when the height of the water level is 10 cm.[4]

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Mark scheme

Marking points from the official mark scheme (Feb/Mar 2017). The mark or mark type is shown in brackets.

[B1] V = (1/12)h³ oe
[M1A1] Attempt differentiation. Allow incorrect notation for M. For A mark accept their letter for volume - but otherwise correct notation. Allow V' dV/dh = 1/4 h² or dh/dV = 4(12V)^(-2/3)
[DM1] Use chain rule correctly with dV/dt = 20. Any equivalent formulation. Accept non-explicit chain rule (or nothing at all) dh/dt = dh/dV * dV/dt
[A1] (dh/dt) = (4/10²) * 20 = 0.8 or equivalent fraction

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