Waste

How many percents are waste from a circular plate with a radius of 1 m from which we cut a square with the highest area?

Result

p =  36.338 %

Solution:

r=1 m u=2r  u=2⋅ r=2⋅ 1=2 m a=u/2=2/2≐2 m≐1.4142 m  S1=π⋅ r2=3.1416⋅ 12≐3.1416 m2 S2=a2=1.41422=2 m2  p=100⋅ S1−S2S1=100⋅ 3.1416−23.1416≐36.338≐36.338%r=1 \ \text{m} \ \\ u=2r \ \\ \ \\ u=2 \cdot \ r=2 \cdot \ 1=2 \ \text{m} \ \\ a=u/\sqrt{ 2 }=2/\sqrt{ 2 } \doteq \sqrt{ 2 } \ \text{m} \doteq 1.4142 \ \text{m} \ \\ \ \\ S_{1}=\pi \cdot \ r^2=3.1416 \cdot \ 1^2 \doteq 3.1416 \ \text{m}^2 \ \\ S_{2}=a^2=1.4142^2=2 \ \text{m}^2 \ \\ \ \\ p=100 \cdot \ \dfrac{ S_{1}-S_{2} }{ S_{1} }=100 \cdot \ \dfrac{ 3.1416-2 }{ 3.1416 } \doteq 36.338 \doteq 36.338 \%

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