Question

Set up an equation of a tangent to the graph of the following function.
Find the altitude of the cylinder with the greatest lateral area which can be inscribed in a sphere of radius B.

Answer 1

Radius of sphere$=B$, let radius of base of cylinder be 'r' and height be 'h'. Then,

$S=2\pi rh$

Now, $r=B\sin \alpha ,h=2B\cos \alpha$ where $\alpha$ be the angle between radius and axis of the cylinder. Then,

$S=2\pi r^2\sin \alpha \cos \alpha$

For maximum, differentiate S w.r.t. $\alpha$ and put equals to $0$

$\frac{dS}{d\alpha }=2\pi B^2[{\cos }^2\alpha -{\sin }^2\alpha ]$

Now, $\frac{dS}{d\alpha }=0\Rightarrow 2\pi B^2[{\cos }^2\alpha -{\sin }^2\alpha ]=0$

$2\pi B^2\ne 0\therefore {\cos }^2\alpha -{\sin }^2\alpha =0\Rightarrow {\tan }^2\alpha =1\Rightarrow \alpha =45°$

Now, $h=2B\cos 45°=\sqrt{2}B$