The volume of a right circular cone which is obtained from a wooden cube of edge 4.2 dm wasting minimum amount of word is :

Options

A. a.19404 cu. dm
B. b.194.04 cu. dm
C. c.19.404 cu. dm
D. d.1940.4 cu. dm

Answer

To find the volume of a right circular cone obtained from a wooden cube with an edge of 4.2 dm, we will follow these steps:

### Step 1: Identify the dimensions of the cube
The edge of the cube is given as 4.2 dm.

### Step 2: Determine the dimensions of the cone
To minimize the amount of wood wasted, the diameter of the cone will be equal to the edge of the cube. Therefore:
- Diameter of the cone = Edge of the cube = 4.2 dm
- Radius of the cone (r) = Diameter / 2 = 4.2 dm / 2 = 2.1 dm
- Height of the cone (h) = Edge of the cube = 4.2 dm

### Step 3: Use the formula for the volume of a cone
The formula for the volume (V) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]

### Step 4: Substitute the values into the formula
Substituting the values we have:
- \( r = 2.1 \) dm
- \( h = 4.2 \) dm
- \( \pi \approx \frac{22}{7} \)

So, the volume becomes:
\[ V = \frac{1}{3} \times \frac{22}{7} \times (2.1)^2 \times 4.2 \]

### Step 5: Calculate \( (2.1)^2 \)
Calculating \( (2.1)^2 \):
\[ (2.1)^2 = 4.41 \]

### Step 6: Substitute back into the volume formula
Now substituting \( (2.1)^2 \) back into the volume formula:
\[ V = \frac{1}{3} \times \frac{22}{7} \times 4.41 \times 4.2 \]

### Step 7: Simplify the expression
First, simplify \( \frac{1}{3} \times 4.41 \):
\[ \frac{4.41}{3} = 1.47 \]

Now substitute this back:
\[ V = \frac{22}{7} \times 1.47 \times 4.2 \]

### Step 8: Calculate \( \frac{22 \times 1.47 \times 4.2}{7} \)
Calculating \( 22 \times 1.47 \):
\[ 22 \times 1.47 = 32.34 \]

Now, multiply by 4.2:
\[ 32.34 \times 4.2 = 135.828 \]

Finally, divide by 7:
\[ V = \frac{135.828}{7} \approx 19.404 \text{ cubic dm} \]

### Final Answer
The volume of the right circular cone is approximately **19.404 cubic dm**.

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