Problems 1112.33 Find the admissible and constrained variations at the pointX= { 0 , 4 }Tfor the following
problem:
Minimizef=x^21 +(x 2 − 1 )^2
subject to
− 2 x^21 +x 2 = 42.34 Find the diameter of an open cylindrical can that will have the maximum volume for a
given surface area,S.
2.35 A rectangular beam is to be cut from a circular log of radiusr. Find the cross-sectional
dimensions of the beam to (a) maximize the cross-sectional area of the beam, and (b)
maximize the perimeter of the beam section.
2.36 Find the dimensions of a straight beam of circular cross section that can be cut from a
conical log of heighthand base radiusrto maximize the volume of the beam.
2.37 The deflection of a rectangular beam is inversely proportional to the width and the cube
of depth. Find the cross-sectional dimensions of a beam, which corresponds to minimum
deflection, that can be cut from a cylindrical log of radiusr.
2.38 A rectangular box of heightaand widthbis placed adjacent to a wall (Fig. 2.12). Find
the length of the shortest ladder that can be made to lean against the wall.
2.39 Show that the right circular cylinder of given surface (including the ends) and maximum
volume is such that its height is equal to the diameter of the base.
2.40 Find the dimensions of a closed cylindrical soft drink can that can hold soft drink of
volumeVfor which the surface area (including the top and bottom) is a minimum.
2.41 An open rectangular box is to be manufactured from a given amount of sheet metal
(areaS). Find the dimensions of the box to maximize the volume.
Figure 2.12 Ladder against a wall.