Two circular pieces of equal radii and maximum areas, touching each other are cut out from a rectangular cardboard of dimensions 14 cm `xx` 7 cm. Find the area of the remaining cardboard. `("Use" pi = (22)/(7))`

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the area of the rectangular cardboard.
The area \( A \) of a rectangle is given by the formula:
\[ A = \text{length} \times \text{width} \]
For our rectangular cardboard, the dimensions are 14 cm and 7 cm.

\[ A = 14 \, \text{cm} \times 7 \, \text{cm} = 98 \, \text{cm}^2 \]

### Step 2: Determine the radius of the circular pieces.
Since the two circular pieces are touching each other and are of equal radius, we can denote the radius of each circle as \( r \). The two circles will fit within the width of the rectangle, which is 7 cm. Therefore, the diameter of the two circles together will be equal to the width of the rectangle.

\[ \text{Diameter} = 2r = 7 \, \text{cm} \]
From this, we can find the radius \( r \):
\[ r = \frac{7}{2} = 3.5 \, \text{cm} \]

### Step 3: Calculate the area of one circular piece.
The area \( A_c \) of a circle is given by the formula:
\[ A_c = \pi r^2 \]
Substituting \( r = 3.5 \, \text{cm} \) and using \( \pi = \frac{22}{7} \):
\[ A_c = \frac{22}{7} \times (3.5)^2 \]
Calculating \( (3.5)^2 \):
\[ (3.5)^2 = 12.25 \]
Now substituting back:
\[ A_c = \frac{22}{7} \times 12.25 = \frac{22 \times 12.25}{7} = \frac{269.5}{7} = 38.5 \, \text{cm}^2 \]

### Step 4: Calculate the total area of the two circular pieces.
Since there are two circular pieces:
\[ \text{Total area of circles} = 2 \times A_c = 2 \times 38.5 = 77 \, \text{cm}^2 \]

### Step 5: Calculate the area of the remaining cardboard.
To find the area of the remaining cardboard after cutting out the two circles, we subtract the total area of the circles from the area of the rectangle:
\[ \text{Area of remaining cardboard} = \text{Area of rectangle} - \text{Total area of circles} \]
\[ \text{Area of remaining cardboard} = 98 \, \text{cm}^2 - 77 \, \text{cm}^2 = 21 \, \text{cm}^2 \]

### Final Answer:
The area of the remaining cardboard is \( 21 \, \text{cm}^2 \).

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