An aerospace engineer is designing a high-precision optical filter for a satellite sensor in the shape of a circular sector. The filter requires a special vacuum sealant along its entire perimeter. The cost of this sealant is £4.50 per millimetre. One prototype filter has a radius of 6 mm and a central angle of 1.2 radians.
(i) Calculate the area of this prototype filter.
(ii) Determine the total cost of the sealant required for this prototype.
The engineer designs a new filter with a fixed area of 25 mm2mm^2mm2.
(i) Show that the cost, £CCC, of the sealant required for this new filter is given by
C=9(25r+r) C = 9\left(\frac{25}{r} + r\right) C=9(r25+r)where rrr is the radius measured in millimetres.
(ii) Find the value of rrr for which the cost of the sealant is minimized, and justify that your answer corresponds to a minimum cost.
317 exam-style questions on OCR A Level Maths 1.5 Trigonometry, covering 1.5.1 Definitions for all arguments, 1.5.2 Sine and cosine rules, 1.5.3 Area of a triangle, 1.5.4 Radian measure (A-level only), 1.5.5 Small angle approximations (A-level only), 1.5.6 Graphs of basic trigonometric functions, 1.5.7 Exact values in radians (A-level only), 1.5.8 Reciprocal and inverse trigonometric ratios (A-level only), 1.5.9 Graphs of reciprocal and inverse functions (A-level only), 1.5.10 Trigonometric identities, 1.5.11 Further trigonometric identities (A-level only), 1.5.12 Double angle and compound angle formulae (A-level only), 1.5.13 Geometrical proofs of formulae (A-level only), 1.5.14 Harmonic form Rcos / Rsin (A-level only), 1.5.15 Trigonometric equations, 1.5.16 Proof involving trigonometric functions (A-level only), and 1.5.17 Trigonometric functions in context. Each one has a worked solution and a mark scheme showing where the marks go.