A sphere of radius 9 cm is dropped into a cylindrical vessel partly filled with water . The radius of the vessel is 12 cm. If the sphere is submerged completely , then the surface of the water rises by :

Options

A. 27.5 cm
B. 27 cm
C. 12 cm
D. 6.75 cm

Answer

To solve the problem of how much the water level rises when a sphere of radius 9 cm is submerged in a cylindrical vessel with a radius of 12 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate the Volume of the Sphere:**
The formula for the volume of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere. Here, \( r = 9 \) cm.
\[ V = \frac{4}{3} \pi (9)^3 \]
First, calculate \( 9^3 \):
\[ 9^3 = 729 \]
Now substitute this value back into the volume formula:
\[ V = \frac{4}{3} \pi (729) = \frac{2916}{3} \pi = 972 \pi \, \text{cm}^3 \]

2. **Set Up the Volume of the Cylinder:**
The volume of the cylinder can be expressed as:
\[ V = \pi r^2 h \]
where \( r \) is the radius of the cylinder and \( h \) is the height of the water rise. Here, \( r = 12 \) cm.
\[ V = \pi (12)^2 h = \pi (144) h = 144 \pi h \, \text{cm}^3 \]

3. **Equate the Volumes:**
Since the volume of the water displaced by the submerged sphere is equal to the volume of the cylinder that the water rises, we can set the two volumes equal to each other:
\[ 972 \pi = 144 \pi h \]

4. **Solve for \( h \):**
We can cancel \( \pi \) from both sides:
\[ 972 = 144 h \]
Now, divide both sides by 144 to find \( h \):
\[ h = \frac{972}{144} \]
Simplifying this gives:
\[ h = 6.75 \, \text{cm} \]

### Conclusion:
The surface of the water rises by **6.75 cm** when the sphere is submerged completely.

---