Hexagonal prism 2

The regular hexagonal prism has a surface of 140 cm2 and height of 5 cm. Calculate its volume.

Correct result:

V =  20.175 cm3

Solution:

h=5 S=140 S=2⋅ 6⋅ S1+6ah S=2⋅ 6⋅ a2⋅ 3/4+6ah 140=3⋅ 3⋅ a2+30a −5.19615242271a2−30a+140=0 D=b2−4ac=3809.84535672  a1=−8.8261406997≐−8.8261 a2=3.05263800781≐3.0526 a=a2=3.0526≐3.0526 S1=a2⋅ 3/4=3.05262⋅ 3/4≐4.0351 S2=6⋅ a⋅ h=6⋅ 3.0526⋅ 5≐91.5791 S3=2⋅ 6⋅ S1+S2=2⋅ 6⋅ 4.0351+91.5791=140 V=S1⋅ h=4.0351⋅ 5=20.175 cm3h=5 \ \\ S=140 \ \\ S=2 \cdot \ 6 \cdot \ S_{1} + 6ah \ \\ S=2 \cdot \ 6 \cdot \ a^2 \cdot \ \sqrt{ 3 }/4 + 6ah \ \\ 140=3 \cdot \ \sqrt{ 3 } \cdot \ a^2 + 30a \ \\ -5.19615242271a_{2}-30a+140=0 \ \\ D=b_{2} - 4ac=3809.84535672 \ \\ \ \\ a_{1}=-8.8261406997 \doteq -8.8261 \ \\ a_{2}=3.05263800781 \doteq 3.0526 \ \\ a=a_{2}=3.0526 \doteq 3.0526 \ \\ S_{1}=a^2 \cdot \ \sqrt{ 3 }/4=3.0526^2 \cdot \ \sqrt{ 3 }/4 \doteq 4.0351 \ \\ S_{2}=6 \cdot \ a \cdot \ h=6 \cdot \ 3.0526 \cdot \ 5 \doteq 91.5791 \ \\ S_{3}=2 \cdot \ 6 \cdot \ S_{1}+S_{2}=2 \cdot \ 6 \cdot \ 4.0351+91.5791=140 \ \\ V=S_{1} \cdot \ h=4.0351 \cdot \ 5=20.175 \ \text{cm}^3

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