Assertion (R) : There are 1000 balls of diameter 0.6 cm which can be formed by melting a solid sphere of radius 3 cm.
Reason (R) : Number of spherical balls = (Volume of Bigger solid sphere)/(Volume of 1 spherical ball)

Options

A. Both A and R are true and R is the correct explanation of A.
B. Both A and R are true and R is not the correct explanation of A.
C. A is true but R is false
D. A is false but R is true

Answer

To determine whether the assertion and reason provided in the question are correct, we will follow these steps:

### Step 1: Calculate the Volume of the Bigger Solid Sphere
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
For the bigger sphere, the radius \( r \) is 3 cm. Thus, we can calculate its volume as follows:
\[ V_{\text{big}} = \frac{4}{3} \pi (3)^3 \]
Calculating \( 3^3 \):
\[ 3^3 = 27 \]
Now substituting back:
\[ V_{\text{big}} = \frac{4}{3} \pi (27) = 36\pi \text{ cm}^3 \]

### Step 2: Calculate the Volume of One Small Sphere
The diameter of the small sphere is given as 0.6 cm, so the radius \( r \) is:
\[ r = \frac{0.6}{2} = 0.3 \text{ cm} \]
Now, we calculate the volume of the small sphere:
\[ V_{\text{small}} = \frac{4}{3} \pi (0.3)^3 \]
Calculating \( 0.3^3 \):
\[ 0.3^3 = 0.027 \]
Now substituting back:
\[ V_{\text{small}} = \frac{4}{3} \pi (0.027) = 0.036\pi \text{ cm}^3 \]

### Step 3: Calculate the Number of Small Spheres
To find the number of small spheres that can be formed from the volume of the bigger sphere, we use the formula:
\[ \text{Number of small spheres} = \frac{V_{\text{big}}}{V_{\text{small}}} \]
Substituting the volumes we calculated:
\[ \text{Number of small spheres} = \frac{36\pi}{0.036\pi} \]
The \( \pi \) cancels out:
\[ \text{Number of small spheres} = \frac{36}{0.036} \]
Calculating \( \frac{36}{0.036} \):
\[ \frac{36}{0.036} = 1000 \]

### Conclusion
The assertion is correct: there are indeed 1000 small spheres formed from melting the larger sphere. The reason provided is also correct, as it accurately describes how to calculate the number of small spheres based on the volumes.

### Final Answer
Both the assertion (A) and the reason (R) are true, and R is the correct explanation of A.

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