A solid metallic sphere of radius 8 cm is melted to form 64 equal small solid spheres. The ratio of the surface area of this sphere to that of a small sphere is

Options

A. `4:1`
B. `1:16`
C. `16:1`
D. `1:4`

Answer

To solve the problem step by step, we need to find the ratio of the surface area of a large sphere to that of a small sphere formed from it.

### Step 1: Calculate the volume of the large sphere.
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
For the large sphere with a radius \( r = 8 \) cm:
\[ V = \frac{4}{3} \pi (8)^3 = \frac{4}{3} \pi (512) = \frac{2048}{3} \pi \text{ cm}^3 \]

### Step 2: Calculate the volume of one small sphere.
Since the large sphere is melted to form 64 small spheres, the volume of one small sphere \( V_s \) is:
\[ V_s = \frac{V}{64} = \frac{\frac{2048}{3} \pi}{64} = \frac{2048}{192} \pi = \frac{64}{3} \pi \text{ cm}^3 \]

### Step 3: Find the radius of one small sphere.
Using the volume formula for a sphere, we can set the volume of one small sphere equal to the formula:
\[ V_s = \frac{4}{3} \pi r_s^3 \]
Substituting the volume of the small sphere:
\[ \frac{64}{3} \pi = \frac{4}{3} \pi r_s^3 \]
We can cancel \( \frac{\pi}{3} \) from both sides:
\[ 64 = 4 r_s^3 \]
Dividing both sides by 4:
\[ r_s^3 = 16 \]
Taking the cube root:
\[ r_s = \sqrt[3]{16} = 2 \text{ cm} \]

### Step 4: Calculate the surface area of the large sphere.
The surface area \( A \) of a sphere is given by:
\[ A = 4 \pi r^2 \]
For the large sphere:
\[ A_L = 4 \pi (8)^2 = 4 \pi (64) = 256 \pi \text{ cm}^2 \]

### Step 5: Calculate the surface area of one small sphere.
Using the radius of the small sphere:
\[ A_s = 4 \pi (r_s)^2 = 4 \pi (2)^2 = 4 \pi (4) = 16 \pi \text{ cm}^2 \]

### Step 6: Find the ratio of the surface area of the large sphere to that of a small sphere.
The ratio \( R \) is given by:
\[ R = \frac{A_L}{A_s} = \frac{256 \pi}{16 \pi} = \frac{256}{16} = 16 \]

### Final Answer:
The ratio of the surface area of the large sphere to that of a small sphere is \( 16:1 \).

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