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 (MEI) A Level Maths 1.7 Trigonometry, covering 1.7.1 Solve right-angled triangles, 1.7.2 Definitions of sin, cos and tan for any angle, 1.7.3 Graphs of sin, cos and tan, 1.7.4 Exact values of trig functions (degrees), 1.7.5 Area of a triangle, 1.7.6 Sine and cosine rules, 1.7.7 Identity tan = sin/cos, 1.7.8 Identity sin^2 + cos^2 = 1, 1.7.9 Solve simple trigonometric equations, 1.7.10 Exact values of trig functions (radians) (A-level only), 1.7.11 Inverse trigonometric functions (A-level only), 1.7.12 Radians and degree conversion (A-level only), 1.7.13 Arc length and area of a sector (A-level only), 1.7.14 Small angle approximations (A-level only), 1.7.15 Sec, cosec and cot functions (A-level only), 1.7.16 Graphs of reciprocal trig functions (A-level only), 1.7.17 Pythagorean identities for sec and cosec (A-level only), 1.7.18 Compound angle formulae (A-level only), 1.7.19 Double angle identities (A-level only), 1.7.20 Expressions for a cos θ ± b sin θ (A-level only), 1.7.21 Use identities to solve equations (A-level only), 1.7.22 Proofs involving trigonometric functions (A-level only), 1.7.23 Trig to solve problems in context (A-level only), and 1.7 Trigonometry. Each one has a worked solution and a mark scheme showing where the marks go.