A cylindrical vessel of height 5 cm and radius 4 cm is completely filled with sand. When this sand is poured out it forms a right circular cone of radius 6 cm. What will be the height of this cone ? (Take ? `= (22)/(7)` )

Options

A. 6.67 cm
B. 2.22 cm
C. 3.33 cm
D. 1.67 cm

Answer

To find the height of the cone formed by the sand poured out from the cylindrical vessel, we will follow these steps:

### Step 1: Calculate the volume of the cylindrical vessel.
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 \( r = 4 \) cm,
- Height \( h = 5 \) cm,
- \( \pi = \frac{22}{7} \).

Substituting the values:
\[ V = \frac{22}{7} \times (4)^2 \times 5 \]
\[ = \frac{22}{7} \times 16 \times 5 \]
\[ = \frac{22 \times 80}{7} = \frac{1760}{7} \text{ cm}^3 \]

### Step 2: Set up the volume of the cone.
The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
Where:
- \( r \) is the radius of the cone,
- \( h \) is the height of the cone.

Given:
- Radius of the cone \( r = 6 \) cm.

Substituting the values:
\[ V = \frac{1}{3} \times \frac{22}{7} \times (6)^2 \times h \]
\[ = \frac{1}{3} \times \frac{22}{7} \times 36 \times h \]
\[ = \frac{792h}{21} = \frac{264h}{7} \text{ cm}^3 \]

### Step 3: Equate the volumes of the cylinder and the cone.
Since the volume of sand in the cylinder equals the volume of the cone formed:
\[ \frac{1760}{7} = \frac{264h}{7} \]

### Step 4: Solve for \( h \).
To eliminate the fraction, we can multiply both sides by 7:
\[ 1760 = 264h \]

Now, solve for \( h \):
\[ h = \frac{1760}{264} \]
\[ h = \frac{1760 \div 88}{264 \div 88} = \frac{20}{3} \text{ cm} \]

### Step 5: Convert to decimal (if needed).
\[ h \approx 6.67 \text{ cm} \]

Thus, the height of the cone is approximately \( 6.67 \) cm.

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