40   Question 14 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ Turn over 41   Question 14 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 14 is 8 marks) 42   15. A 0.8 rad h r r F E D C B Figure 5 A company makes toys for children. Figure 5 shows the design for a solid toy that looks like a piece of cheese. The toy is modelled so that • face ABC is a sector of a circle with radius r cm and centre A • angle BAC = 0.8 radians • faces ABC and DEF are congruent • edges AD, CF and BE are perpendicular to faces ABC and DEF • edges AD, CF and BE have length h cm Given that the volume of the toy is 240 cm3 (a) show that the surface area of the toy, S cm2, is given by S = 0.8r2 + 1680 r making your method clear. (4) Using algebraic differentiation, (b) find the value of r for which S has a stationary point. (4) (c) Prove, by further differentiation, that this value of r gives the minimum surface area of the toy. 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