The `(3)/(4)` th part of a conical vessel of internal radius 5 cm and height 24 cm is full of water. The water emptied into a cylindrical vessel with internal radius 10 cm. Find the height of water in cylindrical vessel.

Answer

To solve the problem step by step, we need to find the height of water in a cylindrical vessel after transferring water from a conical vessel. Here's how we can do it:

### Step 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 of the cone
- \( h \) is the height of the cone

Given:
- Radius \( r = 5 \) cm
- Height \( h = 24 \) cm

Substituting the values into the formula:
\[ V = \frac{1}{3} \pi (5)^2 (24) \]
\[ V = \frac{1}{3} \pi (25)(24) \]
\[ V = \frac{600}{3} \pi \]
\[ V = 200 \pi \text{ cm}^3 \]

### Step 2: Calculate the volume of water in the conical vessel
Since the conical vessel is filled to \( \frac{3}{4} \) of its total volume, the volume of water \( V_w \) is:
\[ V_w = \frac{3}{4} \times 200 \pi \]
\[ V_w = 150 \pi \text{ cm}^3 \]

### Step 3: Set up the volume of the cylindrical vessel
The volume \( V_c \) of a cylinder is given by:
\[ V_c = \pi r^2 h' \]
Where:
- \( r \) is the radius of the base of the cylinder
- \( h' \) is the height of the cylinder

Given:
- Radius of the cylindrical vessel \( r = 10 \) cm

Substituting the radius into the formula:
\[ V_c = \pi (10)^2 h' \]
\[ V_c = 100 \pi h' \]

### Step 4: Equate the volume of water to the volume of the cylindrical vessel
Since all the water from the conical vessel is transferred to the cylindrical vessel, we have:
\[ V_w = V_c \]
\[ 150 \pi = 100 \pi h' \]

### Step 5: Solve for \( h' \)
Dividing both sides by \( \pi \):
\[ 150 = 100 h' \]
Now, solving for \( h' \):
\[ h' = \frac{150}{100} \]
\[ h' = 1.5 \text{ cm} \]

### Conclusion
The height of the water in the cylindrical vessel is \( 1.5 \) cm.

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