Question

the radii of the bases of two right circular solid cones of same height are r1 and r2 respectively. the cones are melted and recasted into a solid sphere of radius R. show that the height of each cone is given by : h= 4R^3/r^2+r^2

Answer 1

volume of the 2 cones = volume of sphere

Answer 2

Proved that,  h = $\frac{4R^3}{(r_1+r_2)^2}$

Given  

To prove that, h = $\frac{4R^3}{r_1^2+r_2^2}$

From the given data,    

         Radii of the two right circular solid cones of same height be $r_1$ and $r_2$.

              Volume of two cone = volume of sphere

              Volume of first cone = $\frac{1}{3}$ π$r_1^2$ h

              Volume of second cone = $\frac{1}{3}$ π$r_2^2$ h

              Volume of two cone =  $\frac{1}{3}$ π$r_1^2$ h + $\frac{1}{3}$ π$r_2^2$ h

               Volume of sphere = $\frac{4}{3}$ π$R^3$

          Volume of two cone = volume of sphere                    

           Volume of first cone + Volume of second cone = volume of sphere  

                                       $\frac{1}{3}$ π$r_1^2$ h + $\frac{1}{3}$ π$r_2^2$ h = $\frac{4}{3}$ π$R^3$

                                       $\frac{1}{3}$ πh ($r_1+r_2$)² = $\frac{4}{3}$ πR³

                                            h ($r_1+r_2$)² =  $\frac{4}{3}$ R³

                                                     h = $\frac{4R^3}{(r_1+r_2)^2}$

                          Hence, proved.

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