A sphere of radius 6 cm is dropped in a right circular cylindrical vessel, partly filled with water. If the sphere is completely submerged, the water level in the vessel rises by 2 cm. What is the diameter of the vessel ? (in cm.)

Options

A. 10
B. 12
C. 16
D. 24

Answer

To find the diameter of the cylindrical vessel when a sphere of radius 6 cm is submerged, we can follow these steps:

### Step 1: Calculate the volume of the sphere
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere. Here, the radius \( r = 6 \) cm.

Substituting the value:
\[ V = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) = 288 \pi \text{ cm}^3 \]

### Step 2: Relate the volume of the sphere to the volume of water displaced
When the sphere is submerged, it displaces an equal volume of water. The volume of water displaced can also be expressed in terms of the cylinder's dimensions. The volume \( V \) of water displaced in a cylindrical vessel is given by:
\[ V = \pi R^2 h \]
where \( R \) is the radius of the cylinder and \( h \) is the height of the water rise. In this case, the height \( h = 2 \) cm.

### Step 3: Set the volumes equal to each other
Since the volume of the sphere is equal to the volume of water displaced:
\[ 288 \pi = \pi R^2 (2) \]

### Step 4: Simplify the equation
We can cancel \( \pi \) from both sides:
\[ 288 = 2 R^2 \]

Now, divide both sides by 2:
\[ R^2 = 144 \]

### Step 5: Solve for the radius \( R \)
Taking the square root of both sides:
\[ R = \sqrt{144} = 12 \text{ cm} \]

### Step 6: Calculate the diameter of the vessel
The diameter \( D \) of the cylindrical vessel is twice the radius:
\[ D = 2R = 2 \times 12 = 24 \text{ cm} \]

### Final Answer
The diameter of the vessel is **24 cm**.

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