Cylinder in Paraboloid Problem: In Figure 8-3w, the parabola y = 9 x2 is rotated about

Chapter 8, Problem 24

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Cylinder in Paraboloid Problem: In Figure 8-3w, the parabola y = 9 x2 is rotated about the y-axis to form a paraboloid. A cylinder is coaxially inscribed in the paraboloid. Figure 8-3w a. Find the radius and height of the cylinder of maximum volume. Justify your answer. b. Find the radius and height of the cylinder of maximum lateral area. c. Find the radius and height of the cylinder of maximum total area (including the ends). Justify your answer. d. Does the maximum-volume cylinder have the same dimensions as either of the maximum-area cylinders in part b or c? e. Does rotating the maximum-area rectangle in produce the maximumvolume cylinder in this problem? f. If the cylinder of maximum volume is inscribed in the paraboloid formed by rotating the parabola y = a2 x2 about the y-axis, does the ratio (cylinder radius): (paraboloid radius) depend in any way on the length of theparaboloid? (That is, does it depend on thevalue of the constant a?) Justify your answer.