How many balls each having diameter 3 cm can be made from a cuboidal lead solid of dimensions `9cmxx11cmxx12cm`?

Options

A. 72
B. 96
C. 84
D. 98

Answer

To solve the problem of how many balls, each having a diameter of 3 cm, can be made from a cuboidal lead solid with dimensions 9 cm x 11 cm x 12 cm, we can follow these steps:

### Step 1: Calculate the volume of the cuboidal lead solid.
The volume \( V \) of a cuboid is given by the formula:
\[ V = \text{length} \times \text{breadth} \times \text{height} \]
For our cuboid:
- Length = 9 cm
- Breadth = 11 cm
- Height = 12 cm

So, the volume of the cuboid is:
\[ V = 9 \times 11 \times 12 \]

### Step 2: Calculate the volume of one ball.
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Given that the diameter of the ball is 3 cm, the radius \( r \) is:
\[ r = \frac{3}{2} \text{ cm} \]
Now, substituting the radius into the volume formula:
\[ V = \frac{4}{3} \pi \left(\frac{3}{2}\right)^3 \]

### Step 3: Set up the equation to find the number of balls.
Let \( n \) be the number of balls. The total volume of \( n \) balls will equal the volume of the cuboid:
\[ n \times \frac{4}{3} \pi \left(\frac{3}{2}\right)^3 = 9 \times 11 \times 12 \]

### Step 4: Solve for \( n \).
First, calculate the volume of the cuboid:
\[ V_{\text{cuboid}} = 9 \times 11 \times 12 = 1188 \text{ cm}^3 \]

Next, calculate the volume of one ball:
\[ V_{\text{ball}} = \frac{4}{3} \pi \left(\frac{3}{2}\right)^3 = \frac{4}{3} \pi \left(\frac{27}{8}\right) = \frac{4 \times 27 \pi}{24} = \frac{9 \pi}{2} \text{ cm}^3 \]

Now, substituting these volumes into the equation:
\[ n \times \frac{9 \pi}{2} = 1188 \]

To find \( n \):
\[ n = \frac{1188 \times 2}{9 \pi} \]

### Step 5: Calculate the numerical value of \( n \).
Using \( \pi \approx \frac{22}{7} \):
\[ n = \frac{1188 \times 2}{9 \times \frac{22}{7}} = \frac{2376 \times 7}{198} = \frac{16632}{198} = 84 \]

Thus, the number of balls that can be made is:
\[ \boxed{84} \]