Three solid metallic spheres of diameters 6 cm, 8 cm and 10 cm are melted and recast into a new solid sphere. The diameter of the new sphere is :

Options

A. 4 cm
B. 6 cm
C. 8 cm
D. 12 cm

Answer

To find the diameter of the new solid sphere formed by melting three metallic spheres of diameters 6 cm, 8 cm, and 10 cm, we can follow these steps:

### Step 1: Calculate the radius of each sphere
The radius of a sphere is half of its diameter. Therefore, we can calculate the radius of each sphere as follows:
- Radius of the first sphere (diameter = 6 cm):
\[ r_1 = \frac{6}{2} = 3 \text{ cm} \]
- Radius of the second sphere (diameter = 8 cm):
\[ r_2 = \frac{8}{2} = 4 \text{ cm} \]
- Radius of the third sphere (diameter = 10 cm):
\[ r_3 = \frac{10}{2} = 5 \text{ cm} \]

### Step 2: Calculate the volume of each sphere
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Using this formula, we can calculate the volume of each sphere:
- Volume of the first sphere:
\[ V_1 = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36 \pi \text{ cm}^3 \]
- Volume of the second sphere:
\[ V_2 = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \text{ cm}^3 \]
- Volume of the third sphere:
\[ V_3 = \frac{4}{3} \pi (5)^3 = \frac{4}{3} \pi (125) = \frac{500}{3} \pi \text{ cm}^3 \]

### Step 3: Calculate the total volume of the three spheres
Now, we add the volumes of the three spheres to get the total volume:
\[ V_{total} = V_1 + V_2 + V_3 = 36 \pi + \frac{256}{3} \pi + \frac{500}{3} \pi \]
To add these volumes, we convert \( 36 \pi \) into a fraction:
\[ 36 \pi = \frac{108}{3} \pi \]
Now, we can add:
\[ V_{total} = \frac{108}{3} \pi + \frac{256}{3} \pi + \frac{500}{3} \pi = \frac{864}{3} \pi \text{ cm}^3 \]

### Step 4: Set the total volume equal to the volume of the new sphere
Let the radius of the new sphere be \( R \). The volume of the new sphere is:
\[ V_{new} = \frac{4}{3} \pi R^3 \]
Setting the total volume equal to the volume of the new sphere:
\[ \frac{4}{3} \pi R^3 = \frac{864}{3} \pi \]
Canceling \( \frac{4}{3} \pi \) from both sides:
\[ R^3 = 216 \]

### Step 5: Calculate the radius of the new sphere
Taking the cube root of both sides:
\[ R = \sqrt[3]{216} = 6 \text{ cm} \]

### Step 6: Calculate the diameter of the new sphere
The diameter \( D \) of the new sphere is:
\[ D = 2R = 2 \times 6 = 12 \text{ cm} \]

Thus, the diameter of the new solid sphere is **12 cm**.

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