An octagonal stop sign can be cut from a square sheet of metal, 750mm on a side. To the nearest hundredth, what percentage of the metal square is wasted when the octagon is of maximum size ?
Clearly the maximum area is when four of the sides of the octagon are along the sides of the square.
Thus, that leaves us with four unused triangles, each of side s where
s/√2 + s + s/√2 = 750
s = 750/(1+√2)
The area of each such triangle is thus
(750/(1+√2))^2/2, with a total area of
2(750/(1+√2))^2
The ratio of unused area to total area is thus
2(750/(1+√2))^2
--------------------
750^2
= 6-4√2
= 34.3%
Hmmm. Seems kind of high to me. Better double-check my math.
To find the wasted percentage of the metal square, we need to determine the maximum size of the octagon that can be cut from the square sheet of metal.
Step 1: Calculate the side length of the octagon.
The octagon is inscribed in the square, which means each side of the octagon is equal in length to the side of the square. Given that the sheet of metal is square with a side length of 750mm, the side length of the octagon is also 750mm.
Step 2: Calculate the area of the octagon.
To find the area of a regular octagon, we use the formula:
Area = 2 * (1 + √2) * (side length)^2
Substituting the side length of 750mm into the formula:
Area = 2 * (1 + √2) * (750mm)^2
Step 3: Calculate the area of the metal square.
The area of the metal square is simply the square of its side length:
Area of square = (side length)^2
Substituting the side length of 750mm into the formula:
Area of square = (750mm)^2
Step 4: Calculate the wasted area.
The wasted area is the difference between the area of the square and the area of the octagon:
Wasted area = Area of square - Area of octagon
Step 5: Calculate the wasted percentage.
To find the wasted percentage, we divide the wasted area by the area of the square and multiply by 100:
Wasted percentage = (Wasted area / Area of square) * 100
Now, let's perform the calculations.
Area of the octagon:
Area = 2 * (1 + √2) * (750mm)^2
Area of the square:
Area = (750mm)^2
Wasted area:
Wasted area = Area of square - Area of octagon
Wasted percentage:
Wasted percentage = (Wasted area / Area of square) * 100
By plugging in the given values, we can evaluate the wasted percentage.