A cone of radius 3.5 cm and height 12 cm is completely filled with water. This water is emptied into an empty cylindrical vessel of radius 7 cm. What will be the height of water in this vessel ?

Options

A. 2 cm
B. 0.33 cm
C. 0.5 cm
D. 1 cm

Answer

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

### Step-by-Step Solution:

1. **Calculate 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 \( r = 3.5 \) cm
- Height \( h = 12 \) cm

Substituting the values:
\[ V = \frac{1}{3} \pi (3.5)^2 (12) \]
\[ V = \frac{1}{3} \pi (12.25)(12) \]
\[ V = \frac{1}{3} \pi (147) \]
\[ V = 49 \pi \, \text{cm}^3 \]

2. **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 (which we need to find)

Given:
- Radius \( r = 7 \) cm

The volume of the cylinder when filled with water from the cone will be equal to the volume of the cone:
\[ V = \pi (7)^2 h \]
\[ V = \pi (49) h \]
\[ V = 49 \pi h \, \text{cm}^3 \]

3. **Set the Volumes Equal:**
Since the volume of water from the cone is transferred to the cylinder, we set the volumes equal:
\[ 49 \pi = 49 \pi h \]

4. **Solve for \( h \):**
We can cancel \( 49 \pi \) from both sides (as long as \( \pi \neq 0 \)):
\[ 1 = h \]
Thus, the height of water in the cylindrical vessel is:
\[ h = 1 \, \text{cm} \]

### Final Answer:
The height of water in the cylindrical vessel is **1 cm**.