The sides of three cubes of metal are 30 cm , 40 cm and 50 cm respectively . Find the side of new cube formed by melting these cubes together .

Answer

To find the side of the new cube formed by melting together three cubes with sides of 30 cm, 40 cm, and 50 cm, we will follow these steps:

### Step 1: Calculate the volume of each cube
The volume \( V \) of a cube is given by the formula:
\[ V = \text{side}^3 \]

- For the first cube with side 30 cm:
\[ V_1 = 30^3 = 30 \times 30 \times 30 = 27000 \, \text{cm}^3 \]

- For the second cube with side 40 cm:
\[ V_2 = 40^3 = 40 \times 40 \times 40 = 64000 \, \text{cm}^3 \]

- For the third cube with side 50 cm:
\[ V_3 = 50^3 = 50 \times 50 \times 50 = 125000 \, \text{cm}^3 \]

### Step 2: Calculate the total volume of the three cubes
Now, we will add the volumes of the three cubes to find the total volume \( V_{total} \):
\[ V_{total} = V_1 + V_2 + V_3 = 27000 + 64000 + 125000 \]
Calculating this gives:
\[ V_{total} = 216000 \, \text{cm}^3 \]

### Step 3: Find the side of the new cube
Let the side of the new cube be \( s \). The volume of the new cube can also be expressed as:
\[ V_{new} = s^3 \]
Since the total volume of the melted cubes equals the volume of the new cube, we have:
\[ s^3 = 216000 \]

### Step 4: Calculate the side of the new cube
To find \( s \), we take the cube root of both sides:
\[ s = \sqrt[3]{216000} \]
Calculating the cube root:
\[ s = 60 \, \text{cm} \]

### Final Answer
The side of the new cube formed by melting the three cubes together is:
\[ \boxed{60 \, \text{cm}} \]