There is some water in a cylindrical vessel of radus 6 cm a sphere of radus 2 cm is dropped in to the vessel .Find the increase in the height of water surface.

Answer

To find the increase in the height of the water surface when a sphere is dropped into a cylindrical vessel, we can follow these steps:

### Step 1: Identify the given values
- Radius of the cylindrical vessel (R) = 6 cm
- Radius of the sphere (r) = 2 cm

### Step 2: Calculate the volume of the sphere
The formula for the volume of a sphere is given by:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
Substituting the radius of the sphere:
\[ V_{\text{sphere}} = \frac{4}{3} \pi (2)^3 = \frac{4}{3} \pi (8) = \frac{32}{3} \pi \text{ cm}^3 \]

### Step 3: Set up the equation for the volume of the cylinder
The volume of the cylinder that corresponds to the increase in height (h) when the sphere is submerged is given by:
\[ V_{\text{cylinder}} = \pi R^2 h \]
Substituting the radius of the cylinder:
\[ V_{\text{cylinder}} = \pi (6)^2 h = 36 \pi h \text{ cm}^3 \]

### Step 4: Equate the volumes
Since the volume of the water displaced by the sphere is equal to the volume of the cylinder corresponding to the increase in height:
\[ V_{\text{sphere}} = V_{\text{cylinder}} \]
Thus,
\[ \frac{32}{3} \pi = 36 \pi h \]

### Step 5: Cancel π and solve for h
Dividing both sides by π:
\[ \frac{32}{3} = 36h \]
Now, solving for h:
\[ h = \frac{32}{3 \times 36} = \frac{32}{108} = \frac{8}{27} \text{ cm} \]

### Conclusion
The increase in the height of the water surface when the sphere is dropped into the cylindrical vessel is:
\[ \boxed{\frac{8}{27} \text{ cm}} \]

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