SOLUTION: A square piece of cloth of maximum size is cut from a circular piece, and then a circular piece of maximum size is cut from the square piece. If the radius of the original piece of
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Question 1204506: A square piece of cloth of maximum size is cut from a circular piece, and then a circular piece of maximum size is cut from the square piece. If the radius of the original piece of cloth was 72 cm, what is the amount of cloth, in cm² that was wasted?
A) 2486 pi B) 2576 pi C) 2592 pi D) 1296 pi E) 2660 pi Found 2 solutions by math_tutor2020, greenestamps:Answer by math_tutor2020(3817) (Show Source):
Radius of larger circle = 72 cm
Diagonal of square = 2*72 = 144 cm
Side length of square = 144/sqrt(2) = 72*sqrt(2) cm
Radius of smaller circle = 72*sqrt(2)/2 = 36*sqrt(2) cm
Area of large circle = pi*r^2 = pi*(72)^2 = 5184pi
Area of smaller circle = pi*r^2 = pi*(36*sqrt(2))^2 = 2592pi
Area of wasted material = (Area of large circle) - (Area of small circle)
Area of wasted material = 5184pi - 2592pi
Area of wasted material = 2592pi
It's not a coincidence that the smaller circle area and the amount of wasted material is the same.
In other words, the area of the smaller circle is half that of the larger circle.
Here's scratch work to prove this claim.
r = large circle radius
2r = square diagonal
2r/sqrt(2) = r*sqrt(2) = square side length
r*sqrt(2)/2 = smaller circle radius
smaller circle area = pi*(smaller radius)^2
smaller circle area = pi*(r*sqrt(2)/2)^2
smaller circle area = (1/2)*pi*r^2
smaller circle area = (1/2)*(larger circle area)
It is given that the radius of the larger circle is 72 cm.
The side length of the inscribed square is then 72*sqrt(2) cm.
The radius of the smaller circle is then 36*sqrt(2) cm.
The amount of cloth that is wasted is the difference between the area of the larger circle and the area of the smaller circle.
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