A vessel in the form of a hemisphere surrounded by a cylinder (open at the other end) of same radius is full of liquid of whose volume is 432 `pi` `cm^3`. If water is filled into a level which is 1 cm below the top of vessel the volume of the water is 396 `pi cm^3` The radius of the circular end is

Options

A. 16 cm
B. 36 cm
C. 6 cm
D. 396 cm

Answer

To solve the problem step by step, we will use the information given about the vessel, which consists of a hemisphere and a cylinder.

### Step 1: Understand the Volume of the Vessel
The total volume of the vessel (hemisphere + cylinder) is given as:
\[ V = 432 \pi \, \text{cm}^3 \]

### Step 2: Set Up the Volume Equations
Let the radius of the hemisphere and the cylinder be \( r \) cm, and the height of the cylinder be \( h \) cm. The volume of the cylinder is given by:
\[ V_{cylinder} = \pi r^2 h \]
The volume of the hemisphere is given by:
\[ V_{hemisphere} = \frac{2}{3} \pi r^3 \]
Thus, the total volume can be expressed as:
\[ \pi r^2 h + \frac{2}{3} \pi r^3 = 432 \pi \]

### Step 3: Simplify the Equation
Dividing the entire equation by \( \pi \):
\[ r^2 h + \frac{2}{3} r^3 = 432 \]
This is our **Equation 1**.

### Step 4: Set Up the Second Volume Equation
When the water level is 1 cm below the top of the vessel, the height of the water in the cylinder becomes \( h - 1 \) cm. The volume of water in this case is given as:
\[ V_{water} = 396 \pi \, \text{cm}^3 \]
So we can write:
\[ \pi r^2 (h - 1) + \frac{2}{3} \pi r^3 = 396 \pi \]

### Step 5: Simplify the Second Equation
Dividing by \( \pi \):
\[ r^2 (h - 1) + \frac{2}{3} r^3 = 396 \]
This is our **Equation 2**.

### Step 6: Expand and Rearrange Equation 2
Expanding Equation 2:
\[ r^2 h - r^2 + \frac{2}{3} r^3 = 396 \]

### Step 7: Subtract Equation 2 from Equation 1
Now we will subtract Equation 2 from Equation 1:
\[ (r^2 h + \frac{2}{3} r^3) - (r^2 h - r^2 + \frac{2}{3} r^3) = 432 - 396 \]
This simplifies to:
\[ r^2 = 36 \]

### Step 8: Solve for the Radius
Taking the square root of both sides:
\[ r = 6 \, \text{cm} \]

### Conclusion
The radius of the circular end of the vessel is \( 6 \, \text{cm} \).

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