The height of the cylinder of maximum volume which can be inscribed in a sphere of radius r is:
Answer & explanation
Correct answer: option 3
Let h be the height, x be the base radius, and V be the volume of the cylinder.
$∴r^2=x^2+h^2/4$
$V=πx^2h=πh(r^2-h^2/4)=πr^2h-πh^3/4⇒dV/dh=πr^2-3πh^2/4$
Now $dV/dh=0⇒πr^2-3πh^2/4=0⇒4r^2=3h^2⇒h=2r/\sqrt{3}$