A spherical metal ball of radius 8 cm is melted to make 8 smaller identical balls. The radius of each new ball is cm.

Answer

To find the radius of each smaller identical ball made from the larger spherical metal ball, we can follow these steps:

### Step 1: Calculate the volume of the original spherical ball.
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 original ball, the radius \( r = 8 \) cm. Thus, the volume \( V \) is:
\[ V = \frac{4}{3} \pi (8)^3 \]

### Step 2: Calculate \( (8)^3 \).
Calculating \( (8)^3 \):
\[ (8)^3 = 512 \]

### Step 3: Substitute \( (8)^3 \) into the volume formula.
Now substituting back into the volume formula:
\[ V = \frac{4}{3} \pi (512) = \frac{2048}{3} \pi \text{ cm}^3 \]

### Step 4: Determine the volume of each smaller ball.
Since the original ball is melted to form 8 smaller identical balls, the volume of each smaller ball will be:
\[ \text{Volume of each smaller ball} = \frac{\text{Total Volume}}{8} = \frac{\frac{2048}{3} \pi}{8} \]

### Step 5: Simplify the volume of each smaller ball.
Calculating the volume of each smaller ball:
\[ \text{Volume of each smaller ball} = \frac{2048}{3 \times 8} \pi = \frac{2048}{24} \pi = \frac{256}{3} \pi \text{ cm}^3 \]

### Step 6: Use the volume formula to find the radius of each smaller ball.
Let the radius of each smaller ball be \( r \). Using the volume formula again:
\[ \frac{4}{3} \pi r^3 = \frac{256}{3} \pi \]

### Step 7: Cancel \( \pi \) and solve for \( r^3 \).
Cancelling \( \pi \) from both sides:
\[ \frac{4}{3} r^3 = \frac{256}{3} \]
Multiplying both sides by \( 3 \):
\[ 4 r^3 = 256 \]
Dividing by \( 4 \):
\[ r^3 = 64 \]

### Step 8: Calculate \( r \).
Taking the cube root:
\[ r = \sqrt[3]{64} = 4 \text{ cm} \]

### Final Answer:
The radius of each new smaller ball is \( 4 \) cm.

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