Cube into sphere

The cube has brushed a sphere as large as possible. Determine how much percent was the waste.

Result

p =  47.64 %

Solution:

a=1 r=a/2=1/2=12=0.5  V1=a3=13=1 V2=43⋅ π⋅ r3=43⋅ 3.1416⋅ 0.53≐0.5236  p=100⋅ V1−V2V1=100⋅ 1−0.52361≐47.6401≐47.64a=1 \ \\ r=a/2=1/2=\dfrac{ 1 }{ 2 }=0.5 \ \\ \ \\ V_{ 1 }=a^3=1^3=1 \ \\ V_{ 2 }=\dfrac{ 4 }{ 3 } \cdot \ \pi \cdot \ r^3=\dfrac{ 4 }{ 3 } \cdot \ 3.1416 \cdot \ 0.5^3 \doteq 0.5236 \ \\ \ \\ p=100 \cdot \ \dfrac{ V_{ 1 }-V_{ 2 } }{ V_{ 1 } }=100 \cdot \ \dfrac{ 1-0.5236 }{ 1 } \doteq 47.6401 \doteq 47.64 \\%

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