A sphere of diameter 6 cm is dropped in a right circular cylinderical vessel partly filled with water. The diameter of the cylindrical vessel is 12 cm. If the sphere is just completely submerged in water, then the rise of water level in he cylindrical vessel is

Options

A. 2 cm
B. 1 cm
C. 3 cm
D. 4 cm

Answer

To solve the problem, we need to find the rise in water level in a cylindrical vessel when a sphere is submerged in it. Here are the steps to arrive at the solution:

### Step 1: Find the radius of the sphere
The diameter of the sphere is given as 6 cm. To find the radius, we use the formula:
\[ \text{Radius of the sphere} = \frac{\text{Diameter}}{2} = \frac{6 \, \text{cm}}{2} = 3 \, \text{cm} \]

**Hint:** Remember that the radius is half of the diameter.

### Step 2: Calculate the volume of the sphere
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Substituting the radius we found:
\[ V = \frac{4}{3} \pi (3 \, \text{cm})^3 = \frac{4}{3} \pi (27 \, \text{cm}^3) = 36 \pi \, \text{cm}^3 \]

**Hint:** Use the formula for the volume of a sphere and remember to cube the radius.

### Step 3: Find the radius of the cylindrical vessel
The diameter of the cylindrical vessel is given as 12 cm. Therefore, the radius \( R \) of the cylinder is:
\[ R = \frac{12 \, \text{cm}}{2} = 6 \, \text{cm} \]

**Hint:** Again, the radius is half of the diameter.

### Step 4: Set up the equation for the rise in water level
When the sphere is submerged, it displaces a volume of water equal to its own volume. The volume of water displaced can also be expressed in terms of the rise in height \( h \) of the water in the cylinder:
\[ \text{Volume of displaced water} = \text{Base Area of Cylinder} \times \text{Height Increase} \]
The base area \( A \) of the cylinder is given by:
\[ A = \pi R^2 = \pi (6 \, \text{cm})^2 = 36 \pi \, \text{cm}^2 \]
Thus, the volume of displaced water is:
\[ 36 \pi \, \text{cm}^2 \cdot h \]

**Hint:** The base area of the cylinder can be found using the formula for the area of a circle.

### Step 5: Equate the volumes
Setting the volume of the sphere equal to the volume of displaced water:
\[ 36 \pi \, \text{cm}^3 = 36 \pi \, \text{cm}^2 \cdot h \]
Dividing both sides by \( 36 \pi \):
\[ 1 = h \]

**Hint:** When equating volumes, make sure to simplify by canceling common terms.

### Step 6: Conclusion
The rise in water level \( h \) in the cylindrical vessel is:
\[ h = 1 \, \text{cm} \]

**Final Answer:** The rise of water level in the cylindrical vessel is **1 cm**.

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