In a hollow cylinder, a cone of volume `17pi` fits exactly and remaining portion filled by water. Calculate volume of water.

Options

A. `17pi`
B. `34pi`
C. `51pi`
D. `10pi`

Answer

To solve the problem of finding the volume of water in a hollow cylinder that contains a cone of volume \( 17\pi \), we can follow these steps:

### Step 1: Understand the relationship between the volumes
We know that the volume of the hollow cylinder will be the sum of the volume of the cone and the volume of water. Therefore, if we denote:
- Volume of the cylinder as \( V_c \)
- Volume of the cone as \( V_{cone} \)
- Volume of water as \( V_{water} \)

We can express this relationship as:
\[ V_c = V_{cone} + V_{water} \]

### Step 2: Substitute the known values
From the problem, we know that:
\[ V_{cone} = 17\pi \]
Thus, we can rewrite the equation as:
\[ V_c = 17\pi + V_{water} \]

### Step 3: Express the volume of the cylinder
The volume of a cylinder is given by the formula:
\[ V_c = \pi r^2 h \]
Where \( r \) is the radius and \( h \) is the height of the cylinder. Since the cone fits exactly into the cylinder, it shares the same radius and height.

### Step 4: Express the volume of the cone
The volume of a cone is given by the formula:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]

### Step 5: Relate the volumes
Since the cone fits perfectly into the cylinder, we can relate the volumes of the cylinder and the cone. The volume of the cylinder can be expressed in terms of the volume of the cone:
\[ V_c = 3 \times V_{cone} \]
This is because the volume of the cylinder is three times that of the cone when they share the same base and height.

### Step 6: Calculate the volume of the cylinder
Substituting the volume of the cone into the equation for the volume of the cylinder:
\[ V_c = 3 \times 17\pi = 51\pi \]

### Step 7: Calculate the volume of water
Now, we can find the volume of water by substituting \( V_c \) back into the equation:
\[ V_{water} = V_c - V_{cone} = 51\pi - 17\pi = 34\pi \]

### Final Answer
Thus, the volume of water in the hollow cylinder is:
\[ \boxed{34\pi} \]