Three solid cubes of metal have sides 6 cm, 8 cm and 10 cm, respectively. They are melted to form a new cube. What is the surface area of this new cube?

The correct answer is Option 1: 864 cm².

Step-by-Step Explanation:

Step 1: Calculate the volume of each of the three given cubes.

The formula for the volume of a cube with side length $a$ is:

$V=a^3$

For the three given metallic cubes:

Volume of the first cube ($V_1$) with side $a_1=6\text{cm}$:
$V_1=6^3=216{\text{cm}}^3$

Volume of the second cube ($V_2$) with side $a_2=8\text{cm}$:
$V_2=8^3=512{\text{cm}}^3$

Volume of the third cube ($V_3$) with side $a_3=10\text{cm}$:
$V_3={10}^3=1000{\text{cm}}^3$

Step 2: Find the total volume of the new combined cube.

When the three cubes are melted together, the total volume of the new cube ($V$) is equal to the sum of the volumes of the three individual cubes:

$V=V_1+V_2+V_3$
$V=216+512+1000=1728{\text{cm}}^3$

Step 3: Calculate the side length of the new cube.

Let $A$ be the side length of the new cube. Since $V=A^3$:

$A^3=1728$
$A=\sqrt[3]{1728}=12\text{cm}$

Step 4: Compute the surface area of the new cube.

The total surface area ($S$) of a cube with side length $A$ is given by the formula:

$S=6A^2$

Substituting $A=12\text{cm}$:

$S=6\times {12}^2=6\times 144=864{\text{cm}}^2$

Thus, the total surface area of the new cube is 864 cm².