Question 11 (12 marks) We wish to design a peg to fit a square or a round hole as shown. The peg has three parts: . a square end A on the left (side length 2r cm ) . a solid cylinder B in the middle (radius r cm and length h cm) . a round circular end C on the right (radius 2r cm) The centres of both ends and the axis of the middle cylinder must lie along the line LI. (a) A, C and the curved side of the cylinder B are to be made from sheet plastic. Suppose that the cost in dollars of sheet plastic is given by c(r, h) = "(7,h) "ih +rh, for r > 0, h > 0 where a(r, h) is the area in cm of sheet plastic used to make the peg. Show that c(r, h) = 1+ 4x + + rh. h(b) Find the critical point (r, h) of the cost function and show that it is a local minimum.(c) For any value of r and h, find the mass M (in grams) of the cylinder B, given M (1 +x2 +y ) dr dy