O-LevelAdditional MathematicsDifferentiation and integration4037/21 · May/June 2022 · Q98 Marks

In this question all lengths are in metres. [Figure 9.1] The diagram shows a water container in the shape of a triangular prism. The depth of water in the container is h. The container has length 5. The water in the container forms a prism with a uniform cross-section that is an equilateral triangle of side x.

(a)Show that the volume, V, of the water is given by V = 5√3h² / 3.[4]
(b)Water is pumped into the container at a rate of 0.5 m³ per minute. Find the rate at which the depth of the water is increasing when the depth of the water is 0.1 m.[4]

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Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

[M1] Correct expression connecting x and h e.g. h = √(x² - (x/2)²) oe or cos 30 = h/x oe or 1/2 xh = 1/2 x²sin60
[A1] Must be x = 2h/√3 oe
[M1] V = 1/2 x(their 2h/√3) x sin 60° x 5 or 1/2 xh x their 2h/√3 x 5
[A1] Correct completion to given answer V = 5√3 h²/3
[B1] Correct derivative of V e.g. 10√3h/3
[B1] dh/dt = dV/dt * dh/dV soi (Not dV/dh)
[M1] 0.5 ÷ (10√3/3 * 0.1) oe
[A1] 0.866 (metres per minute) or 0.8660[25...] rot to 4 or more sf Allow √3/2 isw

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