A conical vessel whose internal radius is 10 cm and height 72 cm is full of water . If this water is poured into a cylindrical vessel with internal radius 30 cm, the height of the water level rises in it is :

Options

A. `2""(2)/(3) cm `
B. `3""(2)/(3) cm `
C. `5""(2)/(3) cm `
D. none of these

Answer

To solve the problem, we need to find the height of the water level in a cylindrical vessel after pouring water from a conical vessel. We will use the formula for the volume of a cone and a cylinder.

### Step-by-Step Solution:

1. **Calculate the Volume of the Conical Vessel:**
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 and \( h \) is the height. For the conical vessel, the radius \( r = 10 \) cm and the height \( h = 72 \) cm.
\[ V = \frac{1}{3} \pi (10)^2 (72) \]
\[ V = \frac{1}{3} \pi (100) (72) \]
\[ V = \frac{7200}{3} \pi \]
\[ V = 2400 \pi \text{ cm}^3 \]

2. **Set the Volume Equal to the Volume of the Cylindrical Vessel:**
The volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
For the cylindrical vessel, the radius \( r = 30 \) cm. Let the height of the water level in the cylindrical vessel be \( h \).
\[ V = \pi (30)^2 h \]
\[ V = \pi (900) h \]
\[ V = 900 \pi h \text{ cm}^3 \]

3. **Equate the Volumes:**
Since the volume of water remains the same when poured from the cone to the cylinder:
\[ 2400 \pi = 900 \pi h \]

4. **Cancel \( \pi \) from Both Sides:**
\[ 2400 = 900 h \]

5. **Solve for \( h \):**
\[ h = \frac{2400}{900} \]
\[ h = \frac{240}{90} = \frac{24}{9} = \frac{8}{3} \text{ cm} \]

6. **Convert to Mixed Fraction:**
\[ \frac{8}{3} = 2 \frac{2}{3} \text{ cm} \]

### Final Answer:
The height of the water level that rises in the cylindrical vessel is \( 2 \frac{2}{3} \) cm.