(a) A thin plate of constant density p(x, y) = 1 is to occupy the trian- gular region in the first quadrant of the xy-plane having vertices (0,0), (a, 0) and (a,1). What value of a will minimize the plate's moment of inertia about the origin? (b) i) Find the centroid of the plane region defined by the polar coordinate inequalities 0 < r < a, - a 0 a (0 < a < 7). How does the centroid move as a ? ii) Sketch the region for a = 5 5 and show the centroid in your sketch.