Three spherical balls of radius 2 cm, 4 cm and 6 cm are melted to form a new spherical ball. In this process there is a loss of 25% of the material. What is the radius (in cm) of the new ball?

Options

A. 6
B. 8
C. 12
D. 16

Answer

To solve the problem of finding the radius of the new spherical ball formed by melting three smaller spherical balls, we will follow these steps:

### Step 1: Calculate the Volume of Each Sphere
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 first sphere (radius = 2 cm):**
\[ V_1 = \frac{4}{3} \pi (2)^3 = \frac{4}{3} \pi (8) = \frac{32}{3} \pi \text{ cm}^3 \]

**For the second sphere (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 third sphere (radius = 6 cm):**
\[ V_3 = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) = \frac{864}{3} \pi \text{ cm}^3 \]

### Step 2: Sum the Volumes of the Three Spheres
Now, we sum the volumes of the three spheres:
\[ V_{\text{total}} = V_1 + V_2 + V_3 = \frac{32}{3} \pi + \frac{256}{3} \pi + \frac{864}{3} \pi \]
\[ V_{\text{total}} = \frac{32 + 256 + 864}{3} \pi = \frac{1152}{3} \pi = 384 \pi \text{ cm}^3 \]

### Step 3: Calculate the Volume After Loss
Since there is a loss of 25% of the material, we only retain 75% of the total volume:
\[ V_{\text{new}} = 0.75 \times V_{\text{total}} = 0.75 \times 384 \pi = 288 \pi \text{ cm}^3 \]

### Step 4: Set the Volume of the New Sphere Equal to the Retained Volume
Let the radius of the new sphere be \( R \). The volume of the new sphere is given by:
\[ V_{\text{new}} = \frac{4}{3} \pi R^3 \]
Setting this equal to the retained volume:
\[ \frac{4}{3} \pi R^3 = 288 \pi \]

### Step 5: Solve for \( R^3 \)
We can cancel \( \pi \) from both sides:
\[ \frac{4}{3} R^3 = 288 \]
Multiplying both sides by \( \frac{3}{4} \):
\[ R^3 = 288 \times \frac{3}{4} = 216 \]

### Step 6: Find the Radius \( R \)
Taking the cube root of both sides:
\[ R = \sqrt[3]{216} = 6 \text{ cm} \]

Thus, the radius of the new ball is **6 cm**.

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