Hexagonal prism 2

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

Correct answer:

V =  121.0521 cm³

Step-by-step explanation:

$$\begin{gathered} h=5\text{ cm } \\ S=140{\text{cm}}^2 \\ S=2\cdot 6\cdot S_1+6ah \\ S=2\ast 6\ast a^2\ast sqrt(3)\mathrm{/}4+6\ast a\ast h \\ 140=2\cdot 6\cdot a^2\cdot \sqrt{3}\mathrm{/}4+6\cdot a\cdot 5 \\ -5.19615242271a^2-30a+140=0 \\ 5.19615242271a^2+30a-140=0 \\ p=5.19615242271;q=30;r=-140 \\ D=q^2-4pr=30^2-4\cdot 5.19615242271\cdot (-140)=3809.84535672 \\ D>0 \\ {a}_{1,2}={\displaystyle \frac{-q\pm \sqrt{D}}{2p}}={\displaystyle \frac{-30\pm \sqrt{3809.85}}{10.3923048454}} \\ {a}_{1,2}=-2.88675135\pm 5.93938935376 \\ {a}_1=3.05263800781 \\ {a}_2=-8.8261406997 \\ \text{ Factored form of the equation: } \\ 5.19615242271(a-3.05263800781)(a+8.8261406997)=0 \\ a=a_1=3.0526\doteq 3.0526\text{ cm } \\ {S}_1=a^2\cdot \sqrt{3}\mathrm{/}4=3.0526^2\cdot \sqrt{3}\mathrm{/}4\doteq 4.0351{\text{cm}}^2 \\ {S}_2=6\cdot a\cdot h=6\cdot 3.0526\cdot 5\doteq 91.5791{\text{cm}}^2 \\ P=6\cdot S_1=6\cdot 4.0351\doteq 24.2104{\text{cm}}^2 \\ V=P\cdot h=24.2104\cdot 5\doteq 121.0521=121.0521{\text{cm}}^3 \\ \text{ Verifying Solution: } \\ {S}_3=2\cdot P+S_2=2\cdot 24.2104+91.5791=140{\text{cm}}^2 \end{gathered}$$

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