O-LevelAdditional MathematicsDifferentiation and integration4037/12 · May/June 2016 · Q109 Marks
The diagram shows a badge, made of thin sheet metal, consisting of two semi-circular pieces, centres B and C, each of radius x cm. They are attached to each other by a rectangular piece of thin sheet metal, ABCD, such that AB and CD are the radii of the semi-circular pieces and AD = BC = y cm.
- (i)Given that the area of the badge is 20 cm², show that the perimeter, P cm, of the badge is given by P = 2x + 40/x.[4]
- (ii)Given that x can vary, find the minimum value of P, justifying that this value is a minimum.[5]
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Mark scheme
Marking points from the official mark scheme (May/June 2016). The mark or mark type is shown in brackets.
- [1] B1 Area: 20 = πx² + xy
- [1] M1 for attempt to use perimeter and obtain in terms of x only
- [1] A1 all steps seen, www AG
- [1] M1 for attempt to differentiate
- [1] DM1 for equating to zero and attempt to solve at least as far as x² =
- [1] A1 x = 2√5 allow 4.47, √20
- [1] A1 leading to P = 8√5, allow 17.9
- [1] A1 d²P/dx² = 80/x³, always positive so a minimum
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