A conical iron piece having diameter 28 cm and height 30 cm is totally immersed into the water of a cylindrical vessel, resulting in the rise of water level by 6.4 cm. The diameter , in cm, of the vessel is :

Options

A. `3.5`
B. `(35)/(2)`
C. 32
D. 35

Answer

To solve the problem step by step, we will calculate the diameter of the cylindrical vessel based on the volume of the conical iron piece and the rise in water level.

### Step 1: Calculate the volume of the conical iron piece.

The formula for the volume \( V \) of a cone is given by:

\[ V = \frac{1}{3} \pi r^2 h \]

Where:
- \( r \) is the radius of the base of the cone,
- \( h \) is the height of the cone.

Given:
- Diameter of the cone = 28 cm, so the radius \( r = \frac{28}{2} = 14 \) cm.
- Height of the cone \( h = 30 \) cm.

Substituting these values into the volume formula:

\[ V = \frac{1}{3} \pi (14)^2 (30) \]

Calculating \( (14)^2 \):

\[ (14)^2 = 196 \]

Now substituting back into the volume formula:

\[ V = \frac{1}{3} \pi (196)(30) = \frac{5880}{3} \pi = 1960 \pi \text{ cm}^3 \]

### Step 2: Calculate the volume of water displaced in the cylindrical vessel.

The volume of water displaced by the cone is equal to the volume of the cone. The rise in water level in the cylindrical vessel is given as 6.4 cm.

The formula for the volume \( V \) of a cylinder is given by:

\[ V = \pi r^2 h \]

Where:
- \( r \) is the radius of the base of the cylinder,
- \( h \) is the height of the cylinder (which is the rise in water level).

Let the radius of the cylindrical vessel be \( R \). The height \( h = 6.4 \) cm.

Thus, the volume of water displaced is:

\[ V = \pi R^2 (6.4) \]

### Step 3: Set the volumes equal to each other.

Since the volume of the cone is equal to the volume of the water displaced:

\[ 1960 \pi = \pi R^2 (6.4) \]

We can cancel \( \pi \) from both sides:

\[ 1960 = R^2 (6.4) \]

### Step 4: Solve for \( R^2 \).

Rearranging the equation gives:

\[ R^2 = \frac{1960}{6.4} \]

Calculating \( \frac{1960}{6.4} \):

\[ R^2 = 306.25 \]

### Step 5: Calculate \( R \).

Taking the square root of both sides:

\[ R = \sqrt{306.25} = 17.5 \text{ cm} \]

### Step 6: Calculate the diameter of the cylindrical vessel.

The diameter \( D \) of the cylindrical vessel is:

\[ D = 2R = 2 \times 17.5 = 35 \text{ cm} \]

### Final Answer:

The diameter of the vessel is **35 cm**.

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