Find the number of lead balls, each with a diameter of 3 cm, that can be made from a sphere of diameter 21 cm.

Options

A. A)196
B. B)216
C. C)256
D. D)343

Answer

To find the number of lead balls that can be made from a sphere, we need to compare the volume of the sphere with the total volume of the lead balls. Here’s a step-by-step solution:

### Step 1: Calculate the volume of the 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.

Given the diameter of the sphere is 21 cm, the radius \( r \) is:
\[ r = \frac{21}{2} = 10.5 \text{ cm} \]

Now, substituting the radius into the volume formula:
\[ V_{sphere} = \frac{4}{3} \pi (10.5)^3 \]

### Step 2: Calculate \( (10.5)^3 \)
Calculating \( (10.5)^3 \):
\[ (10.5)^3 = 10.5 \times 10.5 \times 10.5 = 1157.625 \]

### Step 3: Substitute back to find the volume of the sphere
Now, substituting back:
\[ V_{sphere} = \frac{4}{3} \pi (1157.625) = \frac{4 \times 1157.625}{3} \pi = \frac{4630.5}{3} \pi \]

### Step 4: Calculate the volume of one lead ball
The diameter of each lead ball is 3 cm, hence the radius \( r \) is:
\[ r = \frac{3}{2} = 1.5 \text{ cm} \]

Using the volume formula for the lead ball:
\[ V_{lead\ ball} = \frac{4}{3} \pi (1.5)^3 \]

### Step 5: Calculate \( (1.5)^3 \)
Calculating \( (1.5)^3 \):
\[ (1.5)^3 = 1.5 \times 1.5 \times 1.5 = 3.375 \]

### Step 6: Substitute back to find the volume of one lead ball
Now substituting back:
\[ V_{lead\ ball} = \frac{4}{3} \pi (3.375) = \frac{4 \times 3.375}{3} \pi = \frac{13.5}{3} \pi = 4.5 \pi \]

### Step 7: Set up the equation for the number of lead balls
Let \( n \) be the number of lead balls. The total volume of \( n \) lead balls is:
\[ n \times V_{lead\ ball} = n \times 4.5 \pi \]

Setting the total volume of the lead balls equal to the volume of the sphere:
\[ n \times 4.5 \pi = \frac{4630.5}{3} \pi \]

### Step 8: Cancel \( \pi \) and solve for \( n \)
Cancelling \( \pi \) from both sides:
\[ n \times 4.5 = \frac{4630.5}{3} \]

Now, solving for \( n \):
\[ n = \frac{4630.5}{3 \times 4.5} \]

Calculating \( 3 \times 4.5 = 13.5 \):
\[ n = \frac{4630.5}{13.5} \]

### Step 9: Perform the division
Calculating \( n \):
\[ n = 343 \]

### Conclusion
Thus, the number of lead balls that can be made from a sphere of diameter 21 cm is \( \boxed{343} \).