(a) Three metallic spherical balls of radii 3 cm , 4cm and 5 cm are melted and reacast in to a big spherical ball find the radius of this big ball.
(b) Three metallic spheres are melted and recast in to a big solid sphere.Find the radius of big solid sphere if the diameter of three metalic sphres are 16 cm . 12 cm and 2 cm.

Answer

To solve the problem step by step, we will break it down into two parts as given in the question.

### Part (a)

**Step 1: Understand the problem**
We have three metallic spherical balls with radii 3 cm, 4 cm, and 5 cm. We need to find the radius of a larger sphere formed by melting these three spheres.

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

**Step 3: Calculate the volumes of the three smaller spheres**
- For the sphere with radius 3 cm:
\[ V_1 = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36\pi \text{ cm}^3 \]
- For the sphere with radius 4 cm:
\[ V_2 = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi (64) = \frac{256}{3}\pi \text{ cm}^3 \]
- For the sphere with radius 5 cm:
\[ V_3 = \frac{4}{3} \pi (5)^3 = \frac{4}{3} \pi (125) = \frac{500}{3}\pi \text{ cm}^3 \]

**Step 4: Find the total volume of the three spheres**
\[ V_{total} = V_1 + V_2 + V_3 = 36\pi + \frac{256}{3}\pi + \frac{500}{3}\pi \]
To add these, convert \( 36\pi \) to a fraction:
\[ 36\pi = \frac{108}{3}\pi \]
Now, add:
\[ V_{total} = \frac{108}{3}\pi + \frac{256}{3}\pi + \frac{500}{3}\pi = \frac{864}{3}\pi \text{ cm}^3 \]

**Step 5: Set the total volume equal to the volume of the larger sphere**
Let the radius of the larger sphere be \( R \):
\[ \frac{4}{3} \pi R^3 = \frac{864}{3}\pi \]
Cancel \( \frac{4}{3} \pi \) from both sides:
\[ R^3 = 216 \]

**Step 6: Calculate the radius of the larger sphere**
\[ R = \sqrt[3]{216} = 6 \text{ cm} \]

### Part (b)

**Step 1: Understand the problem**
We have three metallic spheres with diameters 16 cm, 12 cm, and 2 cm. We need to find the radius of a larger sphere formed by melting these three spheres.

**Step 2: Convert diameters to radii**
- Radius of the first sphere \( R_1 = \frac{16}{2} = 8 \) cm
- Radius of the second sphere \( R_2 = \frac{12}{2} = 6 \) cm
- Radius of the third sphere \( R_3 = \frac{2}{2} = 1 \) cm

**Step 3: Calculate the volumes of the three smaller spheres**
- For the sphere with radius 8 cm:
\[ V_1 = \frac{4}{3} \pi (8)^3 = \frac{4}{3} \pi (512) = \frac{2048}{3}\pi \text{ cm}^3 \]
- For the sphere with radius 6 cm:
\[ V_2 = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) = 288\pi \text{ cm}^3 \]
- For the sphere with radius 1 cm:
\[ V_3 = \frac{4}{3} \pi (1)^3 = \frac{4}{3} \pi (1) = \frac{4}{3}\pi \text{ cm}^3 \]

**Step 4: Find the total volume of the three spheres**
Convert \( 288\pi \) to a fraction:
\[ 288\pi = \frac{864}{3}\pi \]
Now, add:
\[ V_{total} = \frac{2048}{3}\pi + \frac{864}{3}\pi + \frac{4}{3}\pi = \frac{2916}{3}\pi \text{ cm}^3 \]

**Step 5: Set the total volume equal to the volume of the larger sphere**
Let the radius of the larger sphere be \( R \):
\[ \frac{4}{3} \pi R^3 = \frac{2916}{3}\pi \]
Cancel \( \frac{4}{3} \pi \) from both sides:
\[ R^3 = 729 \]

**Step 6: Calculate the radius of the larger sphere**
\[ R = \sqrt[3]{729} = 9 \text{ cm} \]

### Final Answers
(a) The radius of the big ball is **6 cm**.
(b) The radius of the big solid sphere is **9 cm**.