Metallic spheres of radii 7 cm, 9 cm and 11 cm respectively are melted to form a single solid sphere. The radius of the resulting sphere is

Options

A. 10 cm
B. 8.6 cm
C. 11.5 cm
D. 13.39 cm

Answer

To find the radius of the resulting sphere formed by melting three metallic spheres with given radii, we can follow these steps:

### Step 1: Calculate the volumes of the individual spheres
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.

- For the first sphere with radius \( r_1 = 7 \) cm:
\[ V_1 = \frac{4}{3} \pi (7)^3 = \frac{4}{3} \pi (343) = \frac{1372}{3} \pi \, \text{cm}^3 \]

- For the second sphere with radius \( r_2 = 9 \) cm:
\[ V_2 = \frac{4}{3} \pi (9)^3 = \frac{4}{3} \pi (729) = \frac{2916}{3} \pi \, \text{cm}^3 \]

- For the third sphere with radius \( r_3 = 11 \) cm:
\[ V_3 = \frac{4}{3} \pi (11)^3 = \frac{4}{3} \pi (1331) = \frac{5324}{3} \pi \, \text{cm}^3 \]

### Step 2: Add the volumes of the three spheres
Now, we add the volumes of the three spheres to find the total volume:
\[ V_{\text{total}} = V_1 + V_2 + V_3 = \frac{1372}{3} \pi + \frac{2916}{3} \pi + \frac{5324}{3} \pi \]
\[ V_{\text{total}} = \frac{1372 + 2916 + 5324}{3} \pi = \frac{9612}{3} \pi \, \text{cm}^3 \]

### Step 3: Set the total volume equal to the volume of the resulting sphere
Let the radius of the resulting sphere be \( r_4 \). The volume of the resulting sphere can be expressed as:
\[ V_{\text{resulting}} = \frac{4}{3} \pi r_4^3 \]
Setting the total volume equal to the volume of the resulting sphere:
\[ \frac{4}{3} \pi r_4^3 = \frac{9612}{3} \pi \]

### Step 4: Cancel out \( \frac{4}{3} \pi \) from both sides
\[ r_4^3 = \frac{9612}{4} \]
\[ r_4^3 = 2403 \]

### Step 5: Calculate the radius \( r_4 \)
To find \( r_4 \), we take the cube root of 2403:
\[ r_4 = \sqrt[3]{2403} \approx 13.39 \, \text{cm} \]

### Final Answer
The radius of the resulting sphere is approximately \( 13.39 \, \text{cm} \).

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