Sophie dreamt of opening a boutique or becoming an actress or a fashion designer. She thought that she would be offered the position of a manager. After that she would work there and would save enough money for her boutique.
Keeping in mind their lower middle class family background, Jansie discourages Sophie to have such dreams. Jansie is more realistic and practical in life. She knows that big ambitions cost heavy investments which their working-class status cannot afford.
A racing track is built around an elliptical ground whose equation is given by \[ 9x^2 + 16y^2 = 144 \] The width of the track is \(3\) m as shown. Based on the given information answer the following: 
(i) Express \(y\) as a function of \(x\) from the given equation of ellipse.
(ii) Integrate the function obtained in (i) with respect to \(x\).
(iii)(a) Find the area of the region enclosed within the elliptical ground excluding the track using integration.
OR
(iii)(b) Write the coordinates of the points \(P\) and \(Q\) where the outer edge of the track cuts \(x\)-axis and \(y\)-axis in first quadrant and find the area of triangle formed by points \(P,O,Q\).