Metallic spheres of diameters 12 cm, 16 cm and 20 cm respectively, are melted to form a single solid sphere. Find the radius of the resulting sphere.

Answer

To find the radius of the resulting sphere formed by melting three metallic spheres with diameters of 12 cm, 16 cm, and 20 cm, we can follow these steps:

### Step 1: Calculate the radius of each sphere
The radius of a sphere is half of its diameter.

- For the sphere with a diameter of 12 cm:
\[ r_1 = \frac{12}{2} = 6 \text{ cm} \]

- For the sphere with a diameter of 16 cm:
\[ r_2 = \frac{16}{2} = 8 \text{ cm} \]

- For the sphere with a diameter of 20 cm:
\[ r_3 = \frac{20}{2} = 10 \text{ cm} \]

### Step 2: Calculate the volume of each sphere
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]

- Volume of the first sphere:
\[ V_1 = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) = 288 \pi \text{ cm}^3 \]

- Volume of the second sphere:
\[ V_2 = \frac{4}{3} \pi (8)^3 = \frac{4}{3} \pi (512) = \frac{2048}{3} \pi \text{ cm}^3 \]

- Volume of the third sphere:
\[ V_3 = \frac{4}{3} \pi (10)^3 = \frac{4}{3} \pi (1000) = \frac{4000}{3} \pi \text{ cm}^3 \]

### Step 3: Calculate the total volume of the three spheres
Now, we add the volumes of the three spheres:
\[ V_{total} = V_1 + V_2 + V_3 = 288 \pi + \frac{2048}{3} \pi + \frac{4000}{3} \pi \]

To add these volumes, we need a common denominator:
\[ V_{total} = 288 \pi + \left(\frac{2048 + 4000}{3}\right) \pi = 288 \pi + \frac{6048}{3} \pi \]

Convert \( 288 \) to a fraction with a denominator of 3:
\[ 288 = \frac{864}{3} \]
Thus,
\[ V_{total} = \frac{864}{3} \pi + \frac{6048}{3} \pi = \frac{6912}{3} \pi \text{ cm}^3 \]

### Step 4: Set the total volume equal to the volume of the resulting sphere
Let the radius of the resulting sphere be \( r \). The volume of the resulting sphere is:
\[ V = \frac{4}{3} \pi r^3 \]

Setting the total volume equal to the volume of the resulting sphere:
\[ \frac{4}{3} \pi r^3 = \frac{6912}{3} \pi \]

### Step 5: Cancel out \( \frac{4}{3} \pi \) from both sides
\[ r^3 = \frac{6912}{4} = 1728 \]

### Step 6: Calculate the radius
To find \( r \), take the cube root of 1728:
\[ r = \sqrt[3]{1728} = 12 \text{ cm} \]

### Final Answer
The radius of the resulting sphere is **12 cm**.

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