Regular square prism

The volume of a regular square prism is 192 cm³. The size of its base edge and the body height is 1: 3. Calculate the surface of the prism.

Correct answer:

S =  218.9202 cm²

Step-by-step explanation:

$$\begin{gathered} V=192{\text{cm}}^3 \\ a:d=1:3 \\ d=3a \\ V=a^2\cdot h \\ s=\sqrt{2}\cdot a \\ {h}^2=d^2-s^2=d^2-2a^2=9a^2-2a^2 \\ {h}^2=7a^2 \\ h=\sqrt{7}a \\ V=a^2\cdot h \\ V=a^2\cdot \sqrt{7}a \\ V=a^3\cdot \sqrt{7} \\ a=\sqrt[3]{V/\sqrt{7}}=\sqrt[3]{192/\sqrt{7}}\doteq 4.1711\text{cm} \\ h=\sqrt{7}\cdot a=\sqrt{7}\cdot 4.1711\doteq 11.0357\text{cm} \\ S=2\cdot a^2+4\cdot a\cdot h=2\cdot 4.1711^2+4\cdot 4.1711\cdot 11.0357\doteq 218.9202{\text{cm}}^2 \\ \text{ Verifying Solution: } \\ {V}_2=a^2\cdot h=4.1711^2\cdot 11.0357=192{\text{cm}}^3 \\ {d}_2=\sqrt{h^2+a^2+a^2}=\sqrt{11.0357^2+4.1711^2+4.1711^2}\doteq 12.5133\text{cm} \\ {k}_2=d_2/a=12.5133/4.1711=3 \end{gathered}$$

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