The slant side of a right circular cone is l. Show that the semi vertical angle of the cone of maximum volume is `tan^(-1)`(`sqrt 2`)

The base radii of two right circular cones of the same height are in the ratio 2: 3. Find ratio their volumes.

Find the volume of a right circular cone if r = 14 cm and h = 9 cm.

Water is being poured into a vessel in the form of an inverted right circular cone of semi vertical angle 45^@ in such a way that the rate of change of volume at any moment is proportional to the area of the curved surfaces which is wet at that moment.Initially the vessel is full to a height of 1 cm and after 1 second the height is 3 cm.Show that the height of the water after 3 seconds will be 7 cm.

Water is being poured into a vessel in the form of an inverted right circular cone of semi vertical angle 45^@ in such a way that the rate of change of volume at any moment is proportional to the area of the curved surfaces which is wet at that moment.Initially the vessel is full to a height od 2 cm,after 2 seconds the height becomes 10 cm.Show that after 3.5 seconds from the start,the height of water will be 16 cm.

If the sum of the length of the hypotenuse and another side of a right angled triangle is l, show that the area of the triangle is maximum when the angle between them is 60^o .

The total surface area of a right circular cone of slant height 13 cm is 90 pi cm^(2) , find its radius and volume.

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3.