O-LevelAdditional MathematicsDifferentiation and integration4037/22 · Oct/Nov 2013 · Q78 Marks

The diagram shows a box in the shape of a cuboid with a square cross-section of side x cm. The volume of the box is 3500cm³. Four pieces of tape are fastened round the box as shown. The pieces of tape are parallel to the edges of the box. [Figure showing a cuboid with square cross-section 'x cm' and length 'L']

(i)Given that the total length of the four pieces of tape is L cm, show that L = 14x + 7000/x².[3]
(ii)Given that x can vary, find the stationary value of L and determine the nature of this stationary value.[5]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2013). The mark or mark type is shown in brackets.

[B1] l = 3500/x² allow lx² = 3500
[B1] L = 3x+4x+2x+2l RHS 3 terms e.g. 12x + 2x + 2(3500/x²) or better
[DB1ag] Substitute for l and correctly reach L = 14x + 7000/x² Dependent on both previous B marks
[M1A1] dL/dx = 14-14000/x³ M1 either power reduced by one A1 both terms correct
[DM1] Equate dL/dx to 0 and solve Must get xⁿ =
[A1] x=10 Both values
[A1] L=210
[B1] d²y/dx² = 42000/x⁴ and minimum stated Or use of gradient either side of turning point.

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