A conical vessel whose internal base radiusis 18cm and height 60cm is full of a liquid. The entire of the vessel is emptied into a cylindrical vessel with internal radius 15cm. The height ( in cm ) to which the liquid rises in the cylinderical vessel is `:`

Options

A. A)28.8cm
B. B)30.2 cm
C. C)27 cm
D. D)24 cm

Answer

To solve the problem, we need to find the height to which the liquid rises in the cylindrical vessel after being poured from the conical vessel. 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 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 of the base and \( h \) is the height of the cone.

For the conical vessel:
- Radius \( r = 18 \) cm
- Height \( h = 60 \) cm

Substituting these values into the formula:
\[ V = \frac{1}{3} \pi (18)^2 (60) \]
\[ V = \frac{1}{3} \pi (324) (60) \]
\[ V = \frac{1}{3} \pi (19440) \]
\[ V = 6480 \pi \text{ cm}^3 \]

2. **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 base and \( h \) is the height of the cylinder.

For the cylindrical vessel:
- Radius \( r = 15 \) cm
- Height \( h = h \) (unknown)

Substituting the radius into the formula:
\[ V = \pi (15)^2 h \]
\[ V = \pi (225) h \]
\[ V = 225 \pi h \text{ cm}^3 \]

3. **Set the Volumes Equal:**
Since the entire volume of the liquid from the conical vessel is transferred to the cylindrical vessel, we can set the two volumes equal to each other:
\[ 6480 \pi = 225 \pi h \]

4. **Solve for \( h \):**
We can divide both sides by \( \pi \) (since \( \pi \) is common and non-zero):
\[ 6480 = 225 h \]
Now, solve for \( h \):
\[ h = \frac{6480}{225} \]
\[ h = 28.8 \text{ cm} \]

### Final Answer:
The height to which the liquid rises in the cylindrical vessel is \( 28.8 \) cm.

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