A-LevelMathematicsDifferentiation9709/12 · Oct/Nov 2015 · Q36 Marks
[Figure 1] shows an open tank in the shape of a triangular prism. The vertical ends ABE and DCF are identical isosceles triangles. Angle ABE = angle BAE = 30°. The length of AD is 40 cm. The tank is fixed in position with the open top ABCD horizontal. Water is poured into the tank at a constant rate of 200 cm³ s⁻¹. The depth of water, t seconds after filling starts, is h cm (see [Figure 2]).
- (i)Show that, when the depth of water in the tank is h cm, the volume, V cm³, of water in the tank is given by V = (40√3)h².[3]
- (ii)Find the rate at which h is increasing when h = 5.[3]
This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2015). The mark or mark type is shown in brackets.
- [B1] x = h tan 60 or x = h√3 (Any correct unsimplified length)
- [M1] Area = (1/2) × (2x) × h (Correct method for area)
- [A1] V = 40√(3h²) (ag)
- [B1] dV/dh = 80√(3h)
- [B1] dV/dt = 200
- [M1] dh/dt = (dh/dV) × (dV/dt) (M1 must be division, not multiplication)
- [A1] 0.289
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