Metallic spheres of radii 6cm, 8 cm and 10cm, 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 given radii, we will follow these steps:

### Step 1: Calculate the volumes of the individual spheres
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

For the three spheres:
- For the first sphere with radius \( r_1 = 6 \) cm:
\[ V_1 = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) = 288 \pi \, \text{cm}^3 \]

- For the second sphere with radius \( r_2 = 8 \) cm:
\[ V_2 = \frac{4}{3} \pi (8)^3 = \frac{4}{3} \pi (512) = \frac{2048}{3} \pi \, \text{cm}^3 \]

- For the third sphere with radius \( r_3 = 10 \) cm:
\[ V_3 = \frac{4}{3} \pi (10)^3 = \frac{4}{3} \pi (1000) = \frac{4000}{3} \pi \, \text{cm}^3 \]

### Step 2: Find the total volume of the spheres
Now, we can find the total volume \( V \) of the resulting sphere by adding the volumes of the three spheres:
\[ V = V_1 + V_2 + V_3 \]
Substituting the values we calculated:
\[ V = 288 \pi + \frac{2048}{3} \pi + \frac{4000}{3} \pi \]

To add these, we need a common denominator. The common denominator for the fractions is 3:
\[ V = 288 \pi + \left(\frac{2048 + 4000}{3}\right) \pi \]
Calculating the sum inside the parentheses:
\[ 2048 + 4000 = 6048 \]
So,
\[ V = 288 \pi + \frac{6048}{3} \pi = 288 \pi + 2016 \pi = (288 + 2016) \pi = 2304 \pi \, \text{cm}^3 \]

### Step 3: Find the radius of the resulting sphere
The volume of the resulting sphere can also be expressed as:
\[ V = \frac{4}{3} \pi R^3 \]
where \( R \) is the radius of the resulting sphere. Setting the two expressions for volume equal gives:
\[ \frac{4}{3} \pi R^3 = 2304 \pi \]
Dividing both sides by \( \pi \):
\[ \frac{4}{3} R^3 = 2304 \]
Multiplying both sides by \( \frac{3}{4} \):
\[ R^3 = 2304 \times \frac{3}{4} = 1728 \]
Now, taking the cube root of both sides:
\[ R = \sqrt[3]{1728} = 12 \, \text{cm} \]

### Final Answer
The radius of the resulting sphere is \( 12 \) cm.

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