An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

.157

109. Describe the greatest parallelopiped in a given
sphere, and the least parallelopiped about it; and compare
their volumes and surfaces.
110. The surface of a right cone being given, find its
dimensions when the volume is the greatest possible.
111. Given the volume of a right cone, required its
dimensions when the whole surface is the least possible.
112. Determine the rectangular parallelopiped of a given
volume, which has the least possible surface.
113. A certain quantity of material is to be formed into
a spherical segment of given thickness, what will be its dimensions when the included volume is a maximum?
114. A given sphere is to be formed into a solid composed of two equal right cones on opposite sides of a common
base, so that the surface is a minimum: find the dimensions of
the cones and compare their surfaces with that of the sphere.
115. In a right-angled spherical triangle is given an angle
to find the sides, when the difference between the hypothenuse
and the side adjacent to the given angle is a maximum.
116. If two sides of a spherical triangle be given, it is
required to find when the area will be the greatest possible.
117. If the hypothenuse of a right-angled spherical triangle be given, find its dimensions when the sum of the cosines
of the sides is the least possible.
118. If the sum of two sides, and the included angle of
a spherical triangle be given, it is required to determine when
the sum of the remaining angles will be the least possible.
119. Determine that point in the arc of a quadrant from
which two lines being drawn, one to the centre and the other
bisecting the radius, the included angle shall be the greatest
possible.
120. Of all isoperimetrical polygons having the same
number of sides, the greatest is that which is equilateral
required a proof.