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:

$r = 1 \ m \ \\ u = 2r \ \\ \ \\ u = 2 \cdot \ r = 2 \cdot \ 1 = 2 \ m \ \\ a = u/\sqrt{ 2 } = 2/\sqrt{ 2 } = \sqrt{ 2 } \ m \doteq 1.4142 \ m \ \\ \ \\ S_{ 1 } = \pi \cdot \ r^2 = 3.1416 \cdot \ 1^2 \doteq 3.1416 \ m^2 \ \\ S_{ 2 } = a^2 = 1.4142^2 = 2 \ m^2 \ \\ \ \\ p = 100 \cdot \ \dfrac{ S_{ 1 }-S_{ 2 } }{ S_{ 1 } } = 100 \cdot \ \dfrac{ 3.1416-2 }{ 3.1416 } \doteq 36.338 = 36.338 \%$

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