A cylindrical vessel with radius 8 cm and height 6 cm to be made by melting a number of spherical metallic balls of diameter 4 cm. Find the minimum number of balls needed?

Options

A. 34
B. 36
C. 42
D. 38

Answer

To solve the problem of finding the minimum number of spherical metallic balls needed to create a cylindrical vessel, we will follow these steps:

### Step 1: Calculate the Volume of the Cylinder
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi R^2 H \]
Where:
- \( R \) is the radius of the cylinder
- \( H \) is the height of the cylinder

Given:
- Radius \( R = 8 \) cm
- Height \( H = 6 \) cm

Substituting the values:
\[ V = \pi (8)^2 (6) = \pi (64)(6) = 384\pi \, \text{cm}^3 \]

### Step 2: Calculate the Volume of One 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

Given that the diameter of the sphere is 4 cm, the radius \( r \) is:
\[ r = \frac{4}{2} = 2 \, \text{cm} \]

Substituting the value:
\[ V = \frac{4}{3} \pi (2)^3 = \frac{4}{3} \pi (8) = \frac{32}{3} \pi \, \text{cm}^3 \]

### Step 3: Set Up the Equation to Find the Number of Balls
Let \( n \) be the number of spherical balls needed. The total volume of the spherical balls must equal the volume of the cylinder:
\[ n \times \text{Volume of one sphere} = \text{Volume of the cylinder} \]
\[ n \times \frac{32}{3} \pi = 384 \pi \]

### Step 4: Solve for \( n \)
We can cancel \( \pi \) from both sides:
\[ n \times \frac{32}{3} = 384 \]
Now, multiply both sides by \( 3 \):
\[ 32n = 1152 \]
Now, divide both sides by \( 32 \):
\[ n = \frac{1152}{32} = 36 \]

### Conclusion
The minimum number of spherical balls needed is \( n = 36 \).

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