Question

Three cubes whose edges of length 3 cm, 4 cm, and 5 cm are melted together to form a single cube. Find the surface area of the new cube.

Answer 1

Hint: The formula for the volume of the cube of side a is \[{a^3}\]. The surface area of the cube of the side of the length a is \[6{a^2}\]. Find the length of the side of the new cube and find its surface area.

Complete step-by-step answer:

It is given that three cubes of sides of length 3 cm, 4 cm, and 5 cm are melted to form a new cube.

Let us find the length of the side of the new cube.

The formula for the volume of a cube of side a is given as follows:

\[V = {a^3}...........(1)\]

The volume of the cube of length 3 cm is given using equation (1) as follows:

\[{V_1} = {3^3}\]

\[{V_1} = 27c{m^3}.............(2)\]

The volume of the cube of length 4 cm is given using equation (1) as follows:

\[{V_2} = {4^3}\]

\[{V_2} = 64c{m^3}............(3)\]

The volume of the cube of length 5 cm is given using equation (1) as follows:

\[{V_3} = {5^3}\]

\[{V_3} = 125c{m^3}...........(4)\]

The volume of the new cube is the sum of the volumes of the three cubes.

\[V = {V_1} + {V_2} + {V_3}\]

Using equations (2), (3), and (4), we have:

\[V = 27 + 64 + 125\]

\[V = 216c{m^3}\]

From, equation (1), we have:

\[{a^3} = 216\]

We know that the cube of 6 is 216, hence, we have:

\[a = 6cm\]

The surface area of a cube of side a is given as follows:

\[S = 6{a^2}\]

The surface area of cube of side 6 cm is given as follows:

\[S = 6{(6)^2}\]

\[S = 216c{m^3}\]

Hence, the surface area of the new cube is 216 \[c{m^3}\].

Note: Do not add the surface area of the three cubes to find the surface area of the new cube, only the volume is the same in both and not the surface area.