Two spheres of same metal weight 1 Kg and 7 Kg. The radius of the smaller sphere is 3 cm. The two spheres are melted to form a single big sphere. Find the diameter of the new sphere.

Answer

To solve the problem of finding the diameter of the new sphere formed by melting two smaller spheres, we can follow these steps:

### Step 1: Understand the Problem
We have two spheres with masses \( m_1 = 1 \, \text{kg} \) and \( m_2 = 7 \, \text{kg} \). The radius of the smaller sphere (mass \( m_1 \)) is given as \( r_1 = 3 \, \text{cm} \). We need to find the diameter of the new sphere formed by melting these two spheres.

### Step 2: Find the Volume of the Smaller Sphere
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
For the smaller sphere:
\[ V_1 = \frac{4}{3} \pi (r_1)^3 = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36 \pi \, \text{cm}^3 \]

### Step 3: Find the Volume of the Second Sphere
Since both spheres are made of the same metal, we can use the ratio of their masses to find the volume of the second sphere. The relationship between mass and volume for the same material is:
\[ \frac{m_1}{V_1} = \frac{m_2}{V_2} \]
Rearranging gives:
\[ V_2 = \frac{m_2 \cdot V_1}{m_1} \]
Substituting the known values:
\[ V_2 = \frac{7 \cdot (36 \pi)}{1} = 252 \pi \, \text{cm}^3 \]

### Step 4: Find the Total Volume of the New Sphere
The total volume \( V \) of the new sphere formed by melting the two spheres is:
\[ V = V_1 + V_2 = 36 \pi + 252 \pi = 288 \pi \, \text{cm}^3 \]

### Step 5: Find the Radius of the New Sphere
Using the volume formula for a sphere again, we can find the radius \( R \) of the new sphere:
\[ V = \frac{4}{3} \pi R^3 \]
Setting the volumes equal gives:
\[ 288 \pi = \frac{4}{3} \pi R^3 \]
Dividing both sides by \( \pi \):
\[ 288 = \frac{4}{3} R^3 \]
Multiplying both sides by \( \frac{3}{4} \):
\[ R^3 = 288 \cdot \frac{3}{4} = 216 \]
Taking the cube root:
\[ R = \sqrt[3]{216} = 6 \, \text{cm} \]

### Step 6: Find the Diameter of the New Sphere
The diameter \( D \) of the new sphere is given by:
\[ D = 2R = 2 \times 6 = 12 \, \text{cm} \]

### Final Answer
The diameter of the new sphere is \( \boxed{12 \, \text{cm}} \).