A spherical metal of radius 10 cm is melted and made into 1000 smaller spheres of equal sizes. In this process the surface area of the metal is increased by:

Options

A. 1000 times
B. 100 times
C. 10 times
D. None of these

Answer

To solve the problem, we need to find the increase in surface area when a spherical metal of radius 10 cm is melted and formed into 1000 smaller spheres of equal size.

### Step-by-Step Solution:

1. **Calculate the Surface Area of the Original Sphere:**
The formula for the surface area (SA) of a sphere is:
\[ SA = 4\pi r^2 \]
Given the radius \( r = 10 \) cm, we can substitute this into the formula:
\[ SA = 4\pi (10)^2 = 4\pi (100) = 400\pi \text{ cm}^2 \]

2. **Calculate the Volume of the Original Sphere:**
The formula for the volume (V) of a sphere is:
\[ V = \frac{4}{3}\pi r^3 \]
Substituting the radius \( r = 10 \) cm:
\[ V = \frac{4}{3}\pi (10)^3 = \frac{4}{3}\pi (1000) = \frac{4000}{3}\pi \text{ cm}^3 \]

3. **Set Up the Volume Equation for the Smaller Spheres:**
Let the radius of each smaller sphere be \( r_s \). The volume of one smaller sphere is:
\[ V_s = \frac{4}{3}\pi r_s^3 \]
Since there are 1000 smaller spheres, the total volume of the smaller spheres is:
\[ V_{total} = 1000 \times V_s = 1000 \times \frac{4}{3}\pi r_s^3 \]
Setting the volumes equal (since the metal is not lost):
\[ \frac{4000}{3}\pi = 1000 \times \frac{4}{3}\pi r_s^3 \]

4. **Solve for the Radius of the Smaller Spheres:**
Cancel \(\frac{4}{3}\pi\) from both sides:
\[ 4000 = 1000 \times r_s^3 \]
Dividing both sides by 1000:
\[ r_s^3 = 4 \]
Taking the cube root:
\[ r_s = \sqrt[3]{4} \approx 1.5874 \text{ cm} \]

5. **Calculate the Surface Area of One Smaller Sphere:**
Using the radius \( r_s \):
\[ SA_s = 4\pi r_s^2 = 4\pi (1.5874)^2 \approx 4\pi (2.5198) \approx 10.0792\pi \text{ cm}^2 \]

6. **Calculate the Total Surface Area of the 1000 Smaller Spheres:**
\[ SA_{total} = 1000 \times SA_s = 1000 \times 10.0792\pi \approx 10079.2\pi \text{ cm}^2 \]

7. **Find the Increase in Surface Area:**
The increase in surface area is:
\[ \text{Increase} = SA_{total} - SA_{original} = 10079.2\pi - 400\pi = 10079.2\pi - 400\pi = 9680.2\pi \text{ cm}^2 \]

### Final Answer:
The surface area of the metal is increased by approximately \( 9680.2\pi \text{ cm}^2 \).