A circular cylindrical iron piece of radius. 14 cm and height 20 cm is moulded into a ' solid cone with the same radius of the base. What is the height of the cone ?

Options

A. 36 cm
B. 40 cm
C. 60 cm
D. 48 cm

Answer

To find the height of the cone molded from the cylindrical iron piece, we need to use the principle of conservation of volume. The volume of the cylinder will be equal to the volume of the cone since the material is unchanged.

### Step 1: Calculate the volume of the cylinder.
The formula for the volume of a cylinder is given by:
\[ V_{\text{cylinder}} = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.

Given:
- Radius \( r = 14 \) cm
- Height \( h = 20 \) cm

Substituting the values:
\[ V_{\text{cylinder}} = \pi (14)^2 (20) \]
\[ = \pi \times 196 \times 20 \]
\[ = 3920\pi \, \text{cm}^3 \]

### Step 2: Set up the volume of the cone.
The formula for the volume of a cone is given by:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cone.

Since the cone has the same radius as the cylinder:
- Radius \( r = 14 \) cm

Let the height of the cone be \( h_{\text{cone}} \).

Substituting the values into the volume formula for the cone:
\[ V_{\text{cone}} = \frac{1}{3} \pi (14)^2 h_{\text{cone}} \]
\[ = \frac{1}{3} \pi (196) h_{\text{cone}} \]
\[ = \frac{196\pi}{3} h_{\text{cone}} \, \text{cm}^3 \]

### Step 3: Equate the volumes of the cylinder and the cone.
Since the volume of the cylinder is equal to the volume of the cone:
\[ 3920\pi = \frac{196\pi}{3} h_{\text{cone}} \]

### Step 4: Solve for the height of the cone.
We can cancel \( \pi \) from both sides:
\[ 3920 = \frac{196}{3} h_{\text{cone}} \]

Now, multiply both sides by 3 to eliminate the fraction:
\[ 11760 = 196 h_{\text{cone}} \]

Now, divide both sides by 196 to find \( h_{\text{cone}} \):
\[ h_{\text{cone}} = \frac{11760}{196} \]
\[ h_{\text{cone}} = 60 \, \text{cm} \]

### Final Answer:
The height of the cone is \( 60 \) cm.

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