Question

A sector of a circle of radius 10 cm is folded such that it forms into a cone. If the central angle of the sector is\[{144^0}\] then what is the volume of the cone formed? (In $ c{m^3} $ )

A.   \[\dfrac{{704\sqrt 2 }}{{21}}\] 

B.    $ \dfrac{{628\sqrt {11} }}{{11}} $    

C.   $ \dfrac{{576\sqrt {21} }}{{21}} $  

D.   $ \dfrac{{682\sqrt {11} }}{{11}} $  

Answer 1

Hint: In this question, the first thing that you should do is to write down the formula for finding the volume of cones. Volume of a cone is  $ \dfrac{1}{3}\pi {r^2}h $  ; where r is radius of the cone and h is height of the cone and arc length of sector is  $ 2\pi r\dfrac{\theta }{{{{360}^0}}} $  unit ; where r is radius of arc sector and  $ \theta  $  is central angle of sector, finally put the value in the formula to get the answer.     

Complete step-by-step answer:

In this question, it is given that

Radius of sector = 10 cm

And central angle of the sector =  $ {144^0} $  

Slant height of cone =radius of circle form which sector is cut                                                   

 l = 10 cm                                                                                                                        

Arc length of  $ {144^0} $  sector =  $ 2\pi r\dfrac{\theta }{{{{360}^0}}} $  

$  = 2\pi  \times 10 \times \dfrac{{{{144}^0}}}{{{{360}^0}}} $  cm                                                                             

$  = 8\pi  $ cm                                                                                    

Circumference of the base of cone = Arc length of cut sector =  $ 8\pi  $                                                                

$ \because 2\pi r = 8\pi  $  cm                                                                                                        

$  \Rightarrow r = 4 $ cm                                                                                                                                     

In right angle triangle

 \[ \Rightarrow h = \sqrt {{l^2} - {r^2}} \] cm

                                        \[ = \sqrt {{{10}^2} - {4^2}} \] cm

                                        \[ = \sqrt {100 - 16} \] cm

                                        \[ = \sqrt {84} \] cm

                                        \[ = 2\sqrt {21} \] cm

                                 h  $  = 2\sqrt {21}  $  cm                                                                                                                                   Volume of cone   $  =  $   $ \dfrac{1}{3}\pi {r^2}h $  

                                       $  = \dfrac{1}{3} \times \dfrac{{22}}{7} \times 4 \times 4 \times 2\sqrt {21}  $   $ c{m^3} $ 

                                      $  = \dfrac{{704\sqrt {21} }}{{21}} $   $ c{m^3} $  

Hence, the volume of cone is   $ \dfrac{{704\sqrt {21} }}{{21}} $   $ c{m^3} $.

So option A is correct. 

Note: In this type of question, you should know the formula of cone, and the value of  $ \pi  $  is $ \dfrac{{22}}{7} $   or $ 3.14 $ . In the right angle triangle angle is  $ {90^0} $  or  $ \dfrac{\pi }{2} $  .In right angle triangle  $ h = \sqrt {{l^2} - {b^2}}  $  , this is Pythagoras’s theorem where;  h is hypotenuse , l is perpendicular length and b is base of triangle.