A spherical steel ball was silver polished then it was cut into 4 similar pieces. What is ratio of the polished area to the non polished area :

Options

A. `1:1`
B. `1:2`
C. `2:1`
D. can't be determined

Answer

To solve the problem of finding the ratio of the polished area to the non-polished area after a spherical steel ball is silver polished and then cut into four similar pieces, we can follow these steps:

### Step 1: Calculate the Surface Area of the Sphere
The surface area \( A \) of a sphere is given by the formula:
\[ A = 4\pi r^2 \]
where \( r \) is the radius of the sphere.

### Step 2: Determine the Polished Area
Since the entire sphere is polished, the polished area is equal to the surface area of the sphere:
\[ \text{Polished Area} = 4\pi r^2 \]

### Step 3: Understand the Cutting of the Sphere
When the sphere is cut into 4 equal pieces, each piece is a smaller sphere. The radius of each smaller sphere will be \( \frac{r}{2} \) because the volume is divided into four equal parts.

### Step 4: Calculate the Surface Area of One Smaller Sphere
The surface area \( A' \) of one smaller sphere is given by:
\[ A' = 4\pi \left(\frac{r}{2}\right)^2 = 4\pi \left(\frac{r^2}{4}\right) = \pi r^2 \]

### Step 5: Calculate the Total Surface Area of All Four Smaller Spheres
Since there are four smaller spheres, the total surface area of all four pieces is:
\[ \text{Total Surface Area of 4 pieces} = 4 \times A' = 4 \times \pi r^2 = 4\pi r^2 \]

### Step 6: Determine the Non-Polished Area
When the sphere is cut, the non-polished area consists of the areas of the flat circular faces created by the cuts. Each smaller sphere has a circular face with area:
\[ \text{Area of one circular face} = \pi \left(\frac{r}{2}\right)^2 = \pi \left(\frac{r^2}{4}\right) = \frac{\pi r^2}{4} \]
Since there are 4 pieces, the total non-polished area (the flat circular faces) is:
\[ \text{Total Non-Polished Area} = 4 \times \frac{\pi r^2}{4} = \pi r^2 \]

### Step 7: Calculate the Ratio of Polished Area to Non-Polished Area
Now we can find the ratio of the polished area to the non-polished area:
\[ \text{Ratio} = \frac{\text{Polished Area}}{\text{Non-Polished Area}} = \frac{4\pi r^2}{\pi r^2} = 4:1 \]

### Conclusion
Thus, the ratio of the polished area to the non-polished area is:
\[ \text{Final Answer: } 4:1 \]