Regular 4BH

A regular quadrilateral prism has a volume of 864 cm³ and the area of ​​its surface is twice the area of ​​its base. Determine the size of its body diagonal.

Final Answer:

d =  18 cm

Step-by-step explanation:

$$\begin{gathered} V=864{\text{cm}}^3 \\ {S}_2=2S_1 \\ V=S_1\cdot h \\ {S}_1=a^2 \\ {S}_2=4a\cdot h \\ 4ah=2\cdot a^2 \\ 2h=a \\ V=a^{2}h \\ V=a^2(a/2) \\ a=\sqrt[3]{2\cdot V}=\sqrt[3]{2\cdot 864}=12\text{cm} \\ h=a/2=12/2=6\text{cm} \\ d=\sqrt{a^2+a^2+h^2}=\sqrt{12^2+12^2+6^2}=18=18\text{cm} \\ \text{ Verifying Solution: } \\ {V}_2=a^2\cdot h=12^2\cdot 6=864{\text{cm}}^3 \\ {A}_1=a^2=12^2=144{\text{cm}}^2 \\ {A}_2=4\cdot a\cdot h=4\cdot 12\cdot 6=288{\text{cm}}^2 \\ A=2\cdot A_1+A_2=2\cdot 144+288=576{\text{cm}}^2 \end{gathered}$$

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