Find the volume of the largest cylinder that can be inscribed in a sphere of radius, r.
In order to define the problem please observe this two-dimensional diagram…
If we indicate with $\displaystyle r_{c}$ the radius of base and with $\displaystyle h_{c}$ the height of inscribed cone is…
$\displaystyle r_{c} = r \cdot \cos \theta$
$\displaystyle h_{c} = 2\cdot r\cdot \sin \theta$
The volume of cylinder is therefore…
$\displaystyle V= 2\cdot \pi\cdot r^{3}\cdot \sin \theta \cdot \cos \theta$
Taking the derivative respect to $\displaystyle \theta$ we obtain with little amount of work…
$\displaystyle \frac{dV}{d\theta}= 2\cdot \pi\cdot r^{3}\cdot (cos^{2} \theta -\sin^{2} \theta)$ (1)
Now we have to find the value of $\displaystyle \theta$ for which the derivative vanish and chose the value which maximises the volume of cone. The (1) vanish for…
$\displaystyle \sin^{2} \theta = \cos^{2} \theta$
… and that happens for $\displaystyle \theta = \frac {\pi}{4}$. The maximum volume of an inscribed cylinder is then…
$\displaystyle V= \pi \cdot r^{3}$
Very easy…
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$