Mystery of stereometrie

Two regular tetrahedrons have surfaces 76 cm² and 171 cm². In what ratio are their volumes?

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

Correct answer:

p =  8/27

Step-by-step explanation:

$$\begin{gathered} S\dots a^2 \\ V\dots a^3 \\ {S}_1=k_1{a}_1^2=76{\text{cm}}^2 \\ {S}_2=k_1{a}_2^2=171{\text{cm}}^2 \\ {a}_1:a_2=\sqrt{S_1}:\sqrt{S_2}=\sqrt{76:171} \\ {V}_1:V_2=(a_1:a_2{)}^3=(76:171{)}^{3/2} \\ {V}_1:V_2=(\frac{4}{9}{)}^{3/2}={\frac{2}{3}}^3=\frac{8}{27} \\ p=V_1:V_2 \\ p=\frac{8}{27}=0.2963 \end{gathered}$$

Try another example