Question

A solid is made by mounting the cone onto a hemisphere. The total surface area of the solid is 361.1 cm². If the slant height of the cone is 13 cm find the total height of the solid. (π=3.14)​

Answer 1

GIVEN :-

• Solid is made by mounting the cone into a hemisphere

• Total surface area of solid = 361.1 cm²

• Slant height of the cone = 13 cm

TO FIND :-

• Total height of the solid

SOLUTION :-

  C.S.A of the conical part of solid = $\sf\pi rl$

                                                         $=\sf 3.14\times r \times 13 \\\\= 40.82 r \ cm^2$

  C.S.A of the hemispherical part = $\sf 2\pi r^2$

                                                       $=\sf 2 \times 3.14 \times r^2\\\\= 6.28r^2 \ cm^2$

 Total surface area of the solid =   C.S.A of the conical part of solid +

                                                           C.S.A of the hemispherical part

     

       $\implies \sf 361.1 = 40.82r+6.28r^2\\\\\implies 6.28r^2+40.82r-361.1 = 0 \\\\\implies r^2+ 6.5r - 57.5 = 0 \\\\\implies r^2+11.5r-5r-57.5 = 0 \\\\\implies r(r+11.5)-5(r+11.5)=0\\\\\implies (r+11.5)(r-5)=0\\\\\implies \bf r = -11.5 \ , \ r=5$

      Radius cannot be negative .

       So, r = 5

     We know that ,

            $\sf l^2=h^2+r^2$

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

 Total height of the solid = Height of the cone +

                                              Radius of the hemisphere

                                           = 12 + 5

                                            = 17 cm