Find the area of cardboard wasted if a sector of maximum possible side is cut out from a square cardboard of size 24cm.
the sector will be a 1/4 circle of radius 24...
To find the area of cardboard wasted, we need to calculate the area of the square cardboard and subtract the area of the sector that is cut out.
Step 1: Calculate the area of the square cardboard.
The size of the square cardboard is given as 24 cm on each side. The formula to calculate the area of a square is side length squared (A = s^2).
Area of the square cardboard = 24 cm * 24 cm = 576 cm^2
Step 2: Calculate the area of the sector.
Since the sector is not specified, we need to calculate the area of the sector with the maximum possible side.
The maximum possible side of the sector is the length of the diagonal of the square cardboard (which can be determined using the Pythagorean theorem).
Diagonal of the square = √(side^2 + side^2)
Diagonal of the square = √(24^2 + 24^2)
Diagonal of the square = √(576 + 576)
Diagonal of the square = √1152 = 34.04 cm (rounded to two decimal places)
The diagonal of the square represents the arc length of the sector. To find the angle of the sector, divide the arc length by the radius.
Angle (θ) = Arc Length / Radius
Since the radius is half of the diagonal, the radius is 34.04/2 = 17.02 cm.
Angle (θ) = Arc Length / 17.02 cm
Now, we need to calculate the area of the sector using the formula:
Area of sector = (θ/360) * π * radius^2
Area of sector = (θ/360) * π * (17.02 cm)^2
Substitute the value of θ:
Area of sector = (34.04 cm/360) * π * (17.02 cm)^2
Area of sector ≈ 200.42 cm^2 (rounded to two decimal places)
Step 3: Calculate the wasted area.
To find the wasted area, subtract the area of the sector from the area of the square cardboard:
Wasted area = Area of square cardboard - Area of sector
Wasted area = 576 cm^2 - 200.42 cm^2
Wasted area ≈ 375.58 cm^2 (rounded to two decimal places)
Therefore, the area of cardboard wasted is approximately 375.58 cm^2.