Final answer:
To determine the area of the remaining acrylic sheet after cutting out 56 circular buttons, we first calculate the area of the entire rectangle and then subtract the combined area of all the buttons cut out. The area of the rectangle is 900 cm², and the total area of the buttons is approximately 538.79 cm², resulting in a remaining area of approximately 361.21 cm².
Step-by-step explanation:
To find the area of the remaining acrylic sheet after cutting out 56 circular buttons, we need to calculate the area of the sheet before and after the buttons are cut out. First, we calculate the total area of the buttons and then subtract it from the original area of the rectangular sheet.
The area of the rectangle is calculated as:
Area of Rectangle = Length × Width
Therefore:
Area of Rectangle = 36 cm × 25 cm = 900 cm².
The area of one circular button is calculated using the formula for the area of a circle:
Area of Circle = π × (radius)²
So, with the diameter of each button being 3.5 cm, the radius is half that, which is 1.75 cm. Therefore:
Area of one button = π × (1.75 cm)² = 9.62125 cm² approximately.
We multiply the area of one button by the total number of buttons to find the total area of the buttons:
Total area of buttons = Area of one button × Number of buttons
Total area of buttons = 9.62125 cm² × 56 ≈ 538.79 cm².
Finally, we subtract the total area of buttons from the area of the rectangle to find the remaining area:
Remaining area of sheet = Area of Rectangle - Total area of buttons
Remaining area of sheet ≈ 900 cm² - 538.79 cm² ≈ 361.21 cm².
Therefore, the area of the remaining sheet is approximately 361.21 cm².