Three small lead spheres of radii 3 cm, 4 cm and 5 cm respectively, are melted into a single sphere. The diameter of the new sphere is

Options

A. 6 cm
B. 7 cm
C. 8 cm
D. 12 cm

Answer

To find the diameter of the new sphere formed by melting three smaller spheres of radii 3 cm, 4 cm, and 5 cm, we will follow these steps:

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

1. **Volume of the first sphere (radius = 3 cm)**:
\[ V_1 = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi (27) = 36\pi \, \text{cm}^3 \]

2. **Volume of the second sphere (radius = 4 cm)**:
\[ V_2 = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \, \text{cm}^3 \]

3. **Volume of the third sphere (radius = 5 cm)**:
\[ V_3 = \frac{4}{3} \pi (5)^3 = \frac{4}{3} \pi (125) = \frac{500}{3} \pi \, \text{cm}^3 \]

### Step 2: Calculate the total volume of the three spheres.
Now, we will sum the volumes of the three spheres:
\[ V_{\text{total}} = V_1 + V_2 + V_3 \]
Substituting the volumes we calculated:
\[ V_{\text{total}} = 36\pi + \frac{256}{3}\pi + \frac{500}{3}\pi \]
To add these volumes, we need a common denominator. The common denominator is 3:
\[ V_{\text{total}} = \frac{108}{3}\pi + \frac{256}{3}\pi + \frac{500}{3}\pi = \frac{108 + 256 + 500}{3}\pi = \frac{864}{3}\pi = 288\pi \, \text{cm}^3 \]

### Step 3: Find the radius of the new sphere.
The volume of the new sphere is equal to the total volume we just calculated. Let \( R \) be the radius of the new sphere:
\[ V_{\text{new}} = \frac{4}{3} \pi R^3 \]
Setting this equal to the total volume:
\[ \frac{4}{3} \pi R^3 = 288\pi \]
Dividing both sides by \( \pi \):
\[ \frac{4}{3} R^3 = 288 \]
Multiplying both sides by \( \frac{3}{4} \):
\[ R^3 = 288 \times \frac{3}{4} = 216 \]
Taking the cube root of both sides:
\[ R = \sqrt[3]{216} = 6 \, \text{cm} \]

### Step 4: Calculate the diameter of the new sphere.
The diameter \( D \) of a sphere is given by:
\[ D = 2R \]
Substituting the radius we found:
\[ D = 2 \times 6 = 12 \, \text{cm} \]

### Final Answer:
The diameter of the new sphere is **12 cm**.

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