A-LevelMathematicsDifferentiation9709/12 · Oct/Nov 2009 · Q78 Marks
A piece of wire of length 50 cm is bent to form the perimeter of a sector POQ of a circle. The radius of the circle is r cm and the angle POQ is θ radians (see diagram). [Figure X.X]
- (i)Express θ in terms of r and show that the area, A cm², of the sector is given by A = 25r – r².[4]
- (ii)Given that r can vary, find the stationary value of A and determine its nature.[4]
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Mark scheme
Marking points from the official mark scheme (Oct/Nov 2009). The mark or mark type is shown in brackets.
- [1] Must use s = rθ and link with perimeter (2r + rθ = 50)
- [1] θ = (50-2r)/r (co)
- [1] Used with A = ½r²θ as f(r)
- [1] A = 25r - r² (answer given)
- [1] dA/dr = 25-2r (co)
- [1] r = 12.5 (sets differential to 0 + solution)
- [1] A = 156.25 (co)
- [1] 2nd differential negative -> Maximum (Could be quoted directly from quadratic)
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