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.   $${\displaystyle \frac{704\sqrt{2}}{21}}$$ 

B.    ${\displaystyle \frac{628\sqrt{11}}{11}}$    

C.   ${\displaystyle \frac{576\sqrt{21}}{21}}$  

D.   ${\displaystyle \frac{682\sqrt{11}}{11}}$  

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  ${\displaystyle \frac{1}{3}}\pi r^{2}h$  ; where r is radius of the cone and h is height of the cone and arc length of sector is  $2\pi r{\displaystyle \frac{\theta }{{360}^0}}$  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 =  $2\pi r{\displaystyle \frac{\theta }{{360}^0}}$  

$=2\pi \times 10\times {\displaystyle \frac{{144}^0}{{360}^0}}$  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   $=$   ${\displaystyle \frac{1}{3}}\pi r^{2}h$  

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

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

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

So option A is correct. 

Note: In this type of question, you should know the formula of cone, and the value of  $\pi$  is ${\displaystyle \frac{22}{7}}$   or $3.14$ . In the right angle triangle angle is  ${90}^0$  or  ${\displaystyle \frac{\pi }{2}}$  .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.