Question

the area of base of a cone is 144πcm^2 while its slant height 13cm.this cone is remolded to obtain a soild sphere. the radius of the sphere wii be​

Answer 1

GIVEN :-

• Base area of cone = 144 π cm²

• Slant height of cone , l = 13 cm

• Cone is remolded to obtain a solid sphere .

TO FIND :-

• Radius of the sphere .

SOLUTION :-

   Base area of the cone = $\pi\sf r^2$

           $\hookrightarrow \sf \pi r^ 2 = 144 \pi \\\\\hookrightarrow r^2 = 144\\\\\hookrightarrow r = \sqrt{144} \\\\\hookrightarrow \bf r = 12$

We know that ,

     $\sf l^2 = r^2 + h^2$

   $\hookrightarrow \sf h^2 = l^2-r^2 \\\\\hookrightarrow h =\sqrt{l^2-r^2} \\\\\hookrightarrow h = \sqrt{13^2-12^2} \\\\\hookrightarrow h = \sqrt{169 - 144} \\\\\hookrightarrow h = \sqrt{25} \\\\\hookrightarrow \bf h = 5$

 Cone is remolded to obtain a solid sphere ,

  So Volume of cone = Volume of sphere

  Formula to find the volume of cone = $\sf\dfrac{1}{3}\pi r^2h$

  Formula to find the volume of sphere = $\sf\dfrac{4}{3}\pi r^3$

        $\hookrightarrow \sf \dfrac{1}{3}\pi r^2h = \dfrac{4}{3} \pi r^3\\\\\hookrightarrow \dfrac{1}{3} \pi \times 12 \times 12 \times 5 = \dfrac{4}{3} \pi r^3\\\\\hookrightarrow 240 = \dfrac{4}{3}r^3\\\\\hookrightarrow r^3 = \dfrac{240\times3}{4}\\\\\hookrightarrow r^3 = 180 \\\\\hookrightarrow r =\sqrt[3]{180 } \\\\\hookrightarrow \bf r =5.64 62$

 

    Radius of the sphere = 5.6462 cm