A jogging track in a public park is designed in the form of a closed curve consisting of two semicircles joined by two straight parallel stretches each 70 m long. For aesthetic reasons, the designers now wish to replace the two semicircles by a single ellipse whose major axis coincides with the straight stretches of the original track.
(i) Taking the centre of the proposed ellipse as the origin and the major axis along the x-axis, express its semi-major axis a in terms of the given straight stretch length.
(ii) If the eccentricity of the ellipse is fixed at 0.6 so that the curve looks sufficiently "flat", determine the values of b and c.
(iii) Calculate the total length of the new track correct to the nearest metre, using Ramanujan’s approximation for the perimeter of an ellipse, π[3(a+b)−(3a+b)(a+3b)].