A rectangular sheet of acrylic is 34 cm by 24 cm. From it, 64 circular buttons, each ofdiameter 3.5 cm, have been cut out. Find the area of the remaining sheet.

Answer

To solve the problem step-by-step, we will follow these calculations:

### Step 1: Calculate the area of the rectangular sheet.
The area \( A \) of a rectangle is given by the formula:
\[ A = \text{length} \times \text{breadth} \]
Given:
- Length = 34 cm
- Breadth = 24 cm

Calculating the area:
\[ A = 34 \, \text{cm} \times 24 \, \text{cm} = 816 \, \text{cm}^2 \]

### Step 2: Calculate the radius of one circular button.
The diameter of each button is given as 3.5 cm. The radius \( r \) is half of the diameter:
\[ r = \frac{\text{diameter}}{2} = \frac{3.5 \, \text{cm}}{2} = 1.75 \, \text{cm} \]

### Step 3: Calculate the area of one circular button.
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Using \( \pi \approx \frac{22}{7} \) and substituting the radius:
\[ A = \frac{22}{7} \times (1.75 \, \text{cm})^2 \]
Calculating \( (1.75)^2 \):
\[ (1.75)^2 = 3.0625 \]
Now substituting back:
\[ A = \frac{22}{7} \times 3.0625 \]
To simplify, convert \( 3.0625 \) to a fraction:
\[ 3.0625 = \frac{49}{16} \quad \text{(since } 1.75 = \frac{7}{4} \text{ and } (1.75)^2 = \left(\frac{7}{4}\right)^2 = \frac{49}{16}\text{)} \]
Now substituting:
\[ A = \frac{22 \times 49}{7 \times 16} = \frac{1078}{112} \, \text{cm}^2 \]

### Step 4: Calculate the total area of 64 buttons.
To find the total area of 64 buttons, multiply the area of one button by 64:
\[ \text{Total Area} = 64 \times \frac{1078}{112} = \frac{64 \times 1078}{112} \]
Calculating:
\[ 64 \div 16 = 4 \quad \text{and} \quad 112 \div 16 = 7 \]
So,
\[ \text{Total Area} = \frac{4 \times 1078}{7} = \frac{4312}{7} \approx 616 \, \text{cm}^2 \]

### Step 5: Calculate the area of the remaining sheet.
To find the area of the remaining sheet, subtract the total area of the buttons from the area of the rectangular sheet:
\[ \text{Remaining Area} = \text{Area of Rectangle} - \text{Total Area of Buttons} \]
\[ \text{Remaining Area} = 816 \, \text{cm}^2 - 616 \, \text{cm}^2 = 200 \, \text{cm}^2 \]

### Final Answer:
The area of the remaining sheet is \( 200 \, \text{cm}^2 \).

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