A rectangular plate is of 6 m breadth and 12 m length. Two apertures of 2 m diameter each and one apertures of 1 m diameter have been made with the help of a gas cutter. What is the area of the remaining portion of the plate?

Options

A. 68.5 sq. m.
B. 62.5 sq m
C. 64.5 sq. m
D. None of these

Answer

To find the area of the remaining portion of the rectangular plate after making the apertures, we will follow these steps:

### Step 1: Calculate the area of the rectangular plate
The area \( A \) of a rectangle is given by the formula:
\[ A = \text{length} \times \text{breadth} \]
Given:
- Length = 12 m
- Breadth = 6 m

Calculating the area:
\[ A = 12 \, \text{m} \times 6 \, \text{m} = 72 \, \text{m}^2 \]

### Step 2: Calculate the area of the circular apertures
We have two circular apertures of 2 m diameter and one aperture of 1 m diameter.

#### For the 2 m diameter apertures:
- Radius \( r_1 = \frac{2}{2} = 1 \, \text{m} \)
- Area of one aperture \( A_1 = \pi r_1^2 = \pi (1)^2 = \pi \, \text{m}^2 \)

Since there are two of these:
\[ \text{Total area of 2 m apertures} = 2 \times A_1 = 2 \pi \, \text{m}^2 \]

#### For the 1 m diameter aperture:
- Radius \( r_2 = \frac{1}{2} = 0.5 \, \text{m} \)
- Area of this aperture \( A_2 = \pi r_2^2 = \pi (0.5)^2 = \pi \times 0.25 = 0.25\pi \, \text{m}^2 \)

### Step 3: Calculate the total area of the apertures
Adding the areas of all apertures:
\[ \text{Total area of apertures} = 2\pi + 0.25\pi = 2.25\pi \, \text{m}^2 \]

### Step 4: Calculate the area of the remaining portion of the plate
Now, we subtract the total area of the apertures from the area of the rectangular plate:
\[ \text{Area of remaining portion} = \text{Area of rectangle} - \text{Total area of apertures} \]
Substituting the values:
\[ \text{Area of remaining portion} = 72 - 2.25\pi \]

Using \( \pi \approx 3.14 \):
\[ \text{Total area of apertures} \approx 2.25 \times 3.14 \approx 7.065 \, \text{m}^2 \]
\[ \text{Area of remaining portion} \approx 72 - 7.065 \approx 64.935 \, \text{m}^2 \]

### Final Answer
The area of the remaining portion of the plate is approximately \( 64.93 \, \text{m}^2 \).

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