Three cubes whose edge measure $$3\ cm, 4\ cm,$$ and $$5\ cm$$ respectively are melted to from a new cube. Find the surface area of the new cube formed.
Correct option is A. $$216\ {cm}^{2}$$
$$\textbf{Step 1: Find volumes of three cubes and add them to get total volume.}$$
$$\text{Given, }$$
$$\text{Side of first cube}=3cm$$
$$\text{Side of second cube}=4cm$$
$$\text{Side of third cube}=5cm$$
$$\text{Volume of first cube}=V_1=3^3cm^3$$ $$[\because \textbf{Volume}=\boldsymbol{\textbf{Side}^3}]$$
$$\text{Volume of second cube}=V_2=4^3cm^3$$ $$[\because \textbf{Volume}=\boldsymbol{\textbf{Side}^3}]$$
$$\text{Volume of third cube}=V_3=5^3cm^3$$ $$[\because \textbf{Volume}=\boldsymbol{\textbf{Side}^3}]$$
$$\text{Total volume }=V_1+V_2+V_3$$
$$\text{Total volume }=(27+64+125)cm^3$$
$$=216cm^3$$
$$\textbf{Step 2: Find the side of new cube, and then find it's surface area.}$$
$$\text{Let, a be the side of new cube}$$
$$\therefore a^3=216cm^3$$ $$[\because\textbf{Volume}=\boldsymbol{\textbf{Side}^3}]$$
$$\Rightarrow a=6cm$$
$$\text{Surface area}=6(6^2)cm^2$$ $$[\because\textbf{Surface area}=\boldsymbol{6\times\textbf{Side}^2}]$$
$$=216cm^2$$
$$\textbf{Hence, surface area of new cube is }\boldsymbol{216 cm^2.}$$