The hypotenuse of a right triangle has one end at the origin and one end on the curve y(x) = xe-4X, with x 2 0 . One of the other two sides is on the x-axis, the other side is parallel to the y-axis. X (a) Express the area of the triangle in terms of x only. A(X) = 2.1-6-4x (b) Find A ' (x) = -2x e -4x + 3xe-4x (c) Find the critical point for x > 0. X X (d) Find the maximum area of the triangle. Give an exact answer. AreaFind the point on the parabola shown below that is closest to the point (4, 0). 1 2 (a) Express the square of the distance between the parabola and the point (4, 0) in terms of x only. Note: Minimizing the square of the distance can be used in this case to avoid working with square roots. S(x) = x*+x- 8x +16 (b) Find the coordinates of the point on the parabola to two decimal places. ( 1.13 x )We want to construct a container in the shape of a cylinder. The container will have a bottom but no top. (a) Find a formula for the surface area of the container in terms of the radius and height of the cylinder. O S = Rrh + 2ncr2 O S = 2rrh + Trz O S = nith + Ter 2 O S = 2rrh + 2Tr2 O S = 2rrh + 2nr b) Suppose the container has a fixed volume V. Express the surface area of the container in terms of the radius only. Your answer will also have V in it, but that will be a parameter. Use upper case V and lower case r. You should also simplify your answer. s ( r ) = 2V + Ter 2 (c) Suppose the material for the bottom costs $12 per square foot and the material for the sides costs $6 per square foot. Express the total cost in terms of the radius only. Again, your answer will also have V in it. 12V c(r ) = - + 12 72 12V (d) Find C'(r) = 2 + 24nr (e) Find the critical point of C(r). X