question_answer

The height of the cylinder of maximum volume inscribed in a sphere of radius ?a? is:

A) \[\frac{3a}{2}\]                                

B)  \[\frac{\sqrt{2}a}{3}\]

C)  \[\frac{a}{\sqrt{3}}\]                                    

D)  \[\frac{2}{\sqrt{3a}}\]

Correct Answer: D

Solution :

Let a  be the raidius and \[h\]the height from figure \[{{r}^{2}}+\frac{{{h}^{2}}}{4}={{a}^{2}}\] \[\therefore \]  \[{{h}^{2}}=4({{a}^{2}}-{{r}^{2}})\] Now \[\upsilon =\pi {{r}^{2}}h=\pi \left( {{a}^{2}}-\frac{{{h}^{2}}}{4} \right)h\]                 \[=\pi \left( {{a}^{2}}h-\frac{{{h}^{2}}}{4} \right)\] \[\therefore \]  \[\frac{d\upsilon }{dh}=\pi \left( {{a}^{2}}-\frac{3{{h}^{2}}}{4} \right)=0\] for maximum or minimum \[\Rightarrow \]\[h=2/\sqrt{3}a\Rightarrow \frac{{{d}^{2}}\upsilon }{d{{h}^{2}}}=-\frac{6h}{4}<0\] \[\therefore \]\[\upsilon \]is maximum when \[h=\frac{2}{\sqrt{3}a}\]