question_answer

An inverted cone of vertical height 12 cm and the radius of base 9 cm contains water to a depth of 4 cm. Find the area of the interior surface of the cone not in contact with the water. [Use\[\pi =22/7\]]           

A)  \[402.12\,c{{m}^{2}}\]   

B)        \[298\,c{{m}^{2}}\]       

C)         \[377.14\,c{{m}^{2}}\]   

D)         \[315\,c{{m}^{2}}\]           

Correct Answer: C

Solution :

Since, \[\Delta \text{ }OAB\tilde{\ }\Delta \text{ }OCD\] \[\frac{AB}{CD}=\frac{OB}{OD}\]             \[\Rightarrow \]            \[\frac{9}{CD}=\frac{12}{4}\] \[\Rightarrow \]            \[CD=\frac{9\times 4}{12}\] \[\Rightarrow \]            \[CD=3\,cm\] Slant height of cone \[(l)=\sqrt{{{12}^{2}}+{{9}^{2}}}=15cm\] Curved surface area of cone \[=\pi rl=135\,\pi \] Slant height of conical part containing water \[({{l}_{1}})=\sqrt{{{4}^{2}}+{{3}^{2}}}=5\,cm\] Curved surface area of conical part containing water \[=\pi {{r}_{1}}{{l}_{1}}=15\pi \] Surface area of cone not in contact with water \[=135\pi -15\pi =377.14c{{m}^{2}}\]