Mystery of stereometry

Two regular tetrahedrons have surfaces 92 cm² and 207 cm². In what ratio are their volumes?

Write as a fraction and as a decimal rounded to 4 decimal places.

Final Answer:

p =  8/27

Step-by-step explanation:

$$\begin{gathered} S\dots a^2 \\ V\dots a^3 \\ {S}_1=k_1{a}_1^2=92{\text{cm}}^2 \\ {S}_2=k_1{a}_2^2=207{\text{cm}}^2 \\ {a}_1:a_2=\sqrt{S_1}:\sqrt{S_2}=\sqrt{92:207} \\ {V}_1:V_2=(a_1:a_2{)}^3=(92:207{)}^{3/2} \\ {V}_1:V_2=(\frac{4}{9}{)}^{3/2}={\frac{2}{3}}^3=\frac{8}{27} \\ p=V_1:V_2 \\ p=\frac{8}{27}=0.2963 \end{gathered}$$

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