A solid metallic sphere of radius 6 cm is melted to form 27 equal small solid spheres. The ratio of the surface area of this sphere to that of a small sphere is

Options

A. `3:1`
B. `9:1`
C. `1:9`
D. `1:3`

Answer

To solve the problem step by step, we will follow the outlined process to find the ratio of the surface area of the large sphere to that of a small sphere.

### Step 1: Calculate the Volume of the Large 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. For the large sphere, the radius \( R = 6 \) cm.

\[ V_{\text{large}} = \frac{4}{3} \pi (6)^3 \]
\[ = \frac{4}{3} \pi (216) \]
\[ = \frac{864}{3} \pi \]
\[ = 288 \pi \text{ cm}^3 \]

### Step 2: Calculate the Volume of One Small Sphere
Since the large sphere is melted to form 27 equal small spheres, the volume of one small sphere \( V_{\text{small}} \) is:

\[ V_{\text{small}} = \frac{V_{\text{large}}}{27} = \frac{288 \pi}{27} \]
\[ = \frac{32 \pi}{3} \text{ cm}^3 \]

### Step 3: Set Up the Volume Equation for the Small Sphere
Using the volume formula for a sphere, we can express the volume of a small sphere in terms of its radius \( r \):

\[ V_{\text{small}} = \frac{4}{3} \pi r^3 \]

Setting the two volume equations equal gives us:

\[ \frac{4}{3} \pi r^3 = \frac{32 \pi}{3} \]

### Step 4: Solve for the Radius of the Small Sphere
We can cancel \( \frac{4}{3} \pi \) from both sides:

\[ r^3 = \frac{32}{4} = 8 \]
\[ r = \sqrt[3]{8} = 2 \text{ cm} \]

### Step 5: Calculate the Surface Areas
The surface area \( A \) of a sphere is given by:

\[ A = 4 \pi R^2 \]

**For the large sphere:**
\[ A_{\text{large}} = 4 \pi (6)^2 = 4 \pi (36) = 144 \pi \text{ cm}^2 \]

**For the small sphere:**
\[ A_{\text{small}} = 4 \pi (2)^2 = 4 \pi (4) = 16 \pi \text{ cm}^2 \]

### Step 6: Find the Ratio of the Surface Areas
Now, we can find the ratio of the surface area of the large sphere to that of a small sphere:

\[ \text{Ratio} = \frac{A_{\text{large}}}{A_{\text{small}}} = \frac{144 \pi}{16 \pi} = \frac{144}{16} = 9 \]

Thus, the ratio of the surface area of the large sphere to that of a small sphere is:

\[ \text{Ratio} = 9:1 \]

### Final Answer
The ratio of the surface area of the large sphere to that of a small sphere is \( 9:1 \).

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