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

Answer

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

### Step 1: Understand the relationship between mass, volume, and density.
We know that the density (d) of a material is given by the formula:
\[ d = \frac{mass}{volume} \]
Since both spheres are made of the same metal, their densities are equal.

### Step 2: Set up the equations for the two spheres.
Let the radius of the smaller sphere be \( r_1 = 3 \, \text{cm} \) and its mass \( m_1 = 1 \, \text{kg} \). Let the radius of the larger sphere be \( r_2 \) and its mass \( m_2 = 7 \, \text{kg} \).

Using the formula for density, we have:
\[ \frac{m_1}{V_1} = \frac{m_2}{V_2} \]
Where \( V_1 = \frac{4}{3} \pi r_1^3 \) and \( V_2 = \frac{4}{3} \pi r_2^3 \).

### Step 3: Substitute the values into the equation.
Substituting the values into the density equation gives:
\[ \frac{1}{\frac{4}{3} \pi (3)^3} = \frac{7}{\frac{4}{3} \pi r_2^3} \]
We can simplify this by canceling out \( \frac{4}{3} \pi \):
\[ \frac{1}{27} = \frac{7}{r_2^3} \]

### Step 4: Cross-multiply to find \( r_2^3 \).
Cross-multiplying gives:
\[ r_2^3 = 7 \times 27 \]
Calculating this gives:
\[ r_2^3 = 189 \]

### Step 5: Find \( r_2 \).
Taking the cube root of both sides:
\[ r_2 = \sqrt[3]{189} \]
Calculating the cube root:
\[ r_2 \approx 6 \, \text{cm} \]

### Step 6: Calculate the volume of the new sphere.
The volume of the new sphere formed by melting both spheres is:
\[ V = V_1 + V_2 = \frac{4}{3} \pi r_1^3 + \frac{4}{3} \pi r_2^3 \]

### Step 7: Find the diameter of the new sphere.
The diameter \( D \) of the new sphere is given by:
\[ D = 2 \times r_2 = 2 \times 6 = 12 \, \text{cm} \]

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

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