A-LevelMathematicsDifferentiation9709/12 · Oct/Nov 2012 · Q35 Marks

The diagram shows a plan for a rectangular park ABCD, in which AB = 40 m and AD = 60 m. Points X and Y lie on BC and CD respectively and AX, XY and YA are paths that surround a triangular playground. The length of DY is xm and the length of XC is 2x m. [Figure 3]

(i)Show that the area, A m², of the playground is given by A = x² - 30x + 1200.[2]
(ii)Given that x can vary, find the minimum area of the playground.[3]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2012). The mark or mark type is shown in brackets.

[M1] Attempts at all areas (e.g., A = Area(ABCD) - Area(ΔABX) - Area(ΔXCY) - Area(ΔADY))
[A1] → A = x² - 30x + 1200. (AG)
[B1] dA/dx = 2x - 30 (or completing the square to (x – 15)² + 975)
[M1] Sets differential to 0 + solution. (dA/dx = 0 when x = 15)
[A1] → A = 975.

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