What part of a ditch 48 m long, 16.5 m broad and 4 m deep can be filled by the earth got by digging a cylindrical tunnel of diameter 4 m and length 56 m?

Options

A. `1/9`
B. `2/9`
C. `7/9`
D. `8/9`

Answer

To solve the problem, we need to calculate the volume of the cylindrical tunnel and the volume of the ditch, and then determine what part of the ditch can be filled with the earth dug out from the tunnel.

### Step 1: Calculate the volume of the cylindrical tunnel
The formula for the volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height (or length in this case).

Given:
- Diameter of the tunnel = 4 m, so the radius \( r = \frac{4}{2} = 2 \) m
- Length of the tunnel \( h = 56 \) m

Now substituting the values into the formula:
\[ V = \pi \times (2)^2 \times 56 \]
Using \( \pi \approx \frac{22}{7} \):
\[ V = \frac{22}{7} \times 4 \times 56 \]
Calculating:
\[ V = \frac{22 \times 4 \times 56}{7} = \frac{4928}{7} \approx 704 \text{ m}^3 \]

### Step 2: Calculate the volume of the ditch
The volume \( V \) of a rectangular prism (ditch) is given by:
\[ V = \text{length} \times \text{breadth} \times \text{depth} \]

Given:
- Length of the ditch = 48 m
- Breadth of the ditch = 16.5 m
- Depth of the ditch = 4 m

Now substituting the values into the formula:
\[ V = 48 \times 16.5 \times 4 \]
Calculating:
\[ V = 48 \times 66 = 3168 \text{ m}^3 \]

### Step 3: Determine the part of the ditch that can be filled
Now, we need to find out what part of the ditch can be filled with the earth dug out from the tunnel. This is calculated by dividing the volume of the tunnel by the volume of the ditch:
\[ \text{Part filled} = \frac{\text{Volume of tunnel}}{\text{Volume of ditch}} = \frac{704}{3168} \]

To simplify:
\[ \frac{704}{3168} = \frac{2}{9} \]

### Final Answer
The part of the ditch that can be filled by the earth dug out from the cylindrical tunnel is \( \frac{2}{9} \).