O-LevelAdditional MathematicsDifferentiation and integration4037/22 · May/June 2015 · Q118 Marks

[Figure 11.1] The diagram shows a cuboid of height h units inside a right pyramid OPQRS of height 8 units and with square base of side 4 units. The base of the cuboid sits on the square base PQRS of the pyramid. The points A, B, C and D are corners of the cuboid and lie on the edges OP, OQ, OR and OS, respectively, of the pyramid OPQRS. The pyramids OPQRS and OABCD are similar.

(i)Find an expression for AD in terms of h and hence show that the volume V of the cuboid is given by V = (h³/4) - 4h² + 16h units³.[4]
(ii)Given that h can vary, find the value of h for which V is a maximum.[4]

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Mark scheme

Marking points from the official mark scheme (May/June 2015). The mark or mark type is shown in brackets.

[M1] (8-h)/8 = AD/4 or 8-h : 8 or 8-h : 8-h soi
[A1] AD = 4(8-h)/8 oe or 4(1-h/8) or 4 - h/2 oe
[M1] h x ( (8-h)/8 x 4)² oe h must be in the numerator of the expression for this mark;
[A1] expand and simplify to h³/4 - 4h² + 16h AG
[B1] their h²/4 - 8h + 16 oe
[M1] their (h²/4 - 8h + 16) = 0 and attempt to solve must be a 3-term quadratic; must be an attempt at a derivative
[A2] h = 8/3 oe only or A1 for h = 8/3 and 8
allow 2.67 or 2.66(6...) rot to 4 or more figs for 8/3

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