An aerospace engineer is designing a solar-collecting aperture for a satellite in the shape of a sector of a circle. A specialized gold-plated conductive gasket must be fitted along the entire perimeter of the aperture to prevent electromagnetic interference. The gasket costs £14.40 £14.40\,£14.40 per metre. One prototype design has a radius of 15 m and a central angle of 0.6 radians.
(i) Determine the area of the solar-collecting surface of this prototype aperture.
(ii) Calculate the total cost of the gasket required for this prototype.
For a different mission, the aperture must be redesigned to have a fixed surface area of 45 m2.
(i) Show that the cost, £C£C£C, of the gasket for this redesigned aperture is given by
C=28.8(45r+r) C = 28.8\left(\frac{45}{r} + r\right) C=28.8(r45+r)where r r\,r is the radius measured in metres.
(ii) Hence, find the value of r r\,r for which the gasket cost is minimized, and justify that your value gives a minimum cost.