A cylinder of radius 13 cm and height 56 cm is to be melted to cast 'n' hemispherical bowls of outer diameter 16 cm and inner diameter 14 cm. Find the value of 'n'?

Options

A. A)56
B. B)72
C. C)65
D. D)84

Answer

To find the value of 'n', we need to compare the volume of the cylinder with the total volume of the 'n' hemispherical bowls. Let's go through the solution step by step.

### 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 \( r = 13 \) cm
- Height \( h = 56 \) cm

Substituting the values:
\[ V = \pi (13)^2 (56) \]
Calculating \( 13^2 \):
\[ 13^2 = 169 \]
Now substituting back:
\[ V = \pi \times 169 \times 56 \]
Calculating \( 169 \times 56 \):
\[ 169 \times 56 = 9464 \]
Thus, the volume of the cylinder is:
\[ V = 9464\pi \, \text{cm}^3 \]

### Step 2: Calculate the volume of one hemispherical bowl
The volume \( V \) of a hemispherical bowl can be calculated using the formula:
\[ V = \frac{2}{3} \pi r^3 \]
Where \( r \) is the radius of the hemisphere. However, since we have the outer and inner diameters, we can find the outer and inner radii:
- Outer diameter = 16 cm → Outer radius \( R = \frac{16}{2} = 8 \) cm
- Inner diameter = 14 cm → Inner radius \( r = \frac{14}{2} = 7 \) cm

The volume of the material of one hemispherical bowl is given by the difference between the outer and inner volumes:
\[ V_{\text{bowl}} = \frac{2}{3} \pi R^3 - \frac{2}{3} \pi r^3 \]
Factoring out \( \frac{2}{3} \pi \):
\[ V_{\text{bowl}} = \frac{2}{3} \pi (R^3 - r^3) \]
Substituting the values:
\[ V_{\text{bowl}} = \frac{2}{3} \pi (8^3 - 7^3) \]
Calculating \( 8^3 \) and \( 7^3 \):
\[ 8^3 = 512, \quad 7^3 = 343 \]
Thus:
\[ R^3 - r^3 = 512 - 343 = 169 \]
Now substituting back:
\[ V_{\text{bowl}} = \frac{2}{3} \pi (169) \]
\[ V_{\text{bowl}} = \frac{338}{3} \pi \, \text{cm}^3 \]

### Step 3: Set up the equation for the total volume of 'n' bowls
Since the volume of the cylinder is equal to the total volume of 'n' hemispherical bowls:
\[ 9464\pi = n \cdot \frac{338}{3} \pi \]
Dividing both sides by \( \pi \):
\[ 9464 = n \cdot \frac{338}{3} \]
Multiplying both sides by 3:
\[ 28392 = n \cdot 338 \]
Now, solving for \( n \):
\[ n = \frac{28392}{338} \]
Calculating \( n \):
\[ n = 84 \]

### Final Answer
The value of \( n \) is \( 84 \).

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