O-LevelAdditional MathematicsDifferentiation and integration4037/12 · May/June 2019 · Q109 Marks

The diagram shows an open container in the shape of a cuboid of width x cm, length 4x cm and height h cm. The volume of the container is 800 cm³. [Figure 10.1]

(i)Show that the external surface area, S cm², of the open container is such that S = 4x² + 2000/x.[4]
(ii)Given that x can vary, find the stationary value of S and determine its nature.[5]

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Mark scheme

Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.

[B1] 800 = 4x²h
[B1] h = 800/(4x²) oe or xh = 800/(4x) oe
[M1] (S=)2hx+8xh+4x² oe Allow if h is substituted at this point
[A1] S = 4x² + 2000/x Leading to AG, must have S = or surface area = at some point and no errors
[B1] dS/dx = 8x - 2000/x² For correct differentiation
[M1] When dS/dx = 0, x = ³√250 oe (6.30) For equating to zero and attempt to solve, must get as far as x = (b/a)^(1/3) must be using the form ax + b/x²
[A1] For correct positive x
[A1] S = 476 only
[B1] d²S/dx² = 8 + 4000/x³ d²S/dx² > 0 or 24 so minimum For a correct convincing method, with enough detail to reach a correct conclusion of a minimum. Must be using x = ³√250 oe

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