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.