Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi vertical angle α is one - third that of the cone and the greatest volume of cylinder is
The given right circular cone of fixed height (h) and semi vertical angle ().
Here, a cylinder of radius R and height H is inscribed in the cone.
Then, ∠GAO = α, OG = r, OA = h, OE = R and CE = H.
We get,
r = h tanα
Now, ΔAOG is similar to ΔCEG, we have:
Now, the volume (V) of the cylinder is given by:
Now, if , then,
Now,
Then, by second derivative test, the volume of the cylinder is the greatest when
So, When, H =
Therefore, the height of the cylinder is one - third of the cone when the volume of the cylinder is the greatest.
Now, the maximum volume of the cylinder can be obtained as:
Hence Proved.