Question

Two similar cones have volume of 4 m and 108 m respectively. If the large one has surface area 54 m, find the surface area of the smaller one.​

Answer 1

`\displaystyle\bf\\Explanations:\\\\The~similarity~ratio~of~two~similar~cones=k\\\\k=is~the~ratio~between~2~corresponding~lengths\\\\k=(R_1)/(R_2)=(h_1)/(h_2)\\\\The~ratio~between~2~corresponding~areas~of~similar~cones=k^2\\\\(Area~1)/(Area~2)=k^2\\\\The~ratio~between~the~volumes~of~the~2~similar~cones=k^3\\\\(Volume~1)/(Volume~2)=k^3`

.

`\displaystyle\bf\\Solving:\\\\Volume1=108~m^3\\\\Volume2=4m^3\\\\k^3=(108)/(4)\\\\k^3=27~~\Big|√(~)\\\\k=\sqrt[\b3]{27}\\\\k=3\\\\Area1=54~m^2\\\\(Area~1)/(Area~2)=k^2\\\\(54)/(Area~2)=3^2\\\\(54)/(Area~2)=9\\\\Area2=(54)/(9)\\\\\boxed{\bf Area2=6 m^2}`