Sa se afle \( x,y,z \) stiind ca: \( \frac{x}{1}=\frac{y^2}{8}=\frac{z^3}{4} \)si \( x\cdot{y}\cdot{z}=16 \)
Radu Florica, Simleu Silvaniei
Concursul "Teodor Topan" - problema 2
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Marius Dragoi
- Thales
- Posts: 126
- Joined: Thu Jan 31, 2008 5:57 pm
- Location: Bucharest
Observam ca \( x \) si \( z \) sunt pozitive.
\( z=2^{\frac{2}{3}} x^{\frac{1}{3}} \) , \( y=2^{\frac{3}{2}} x^{\frac{1}{2}} \)
Din \( x y z=16 \) si relatiile precedente obtinem: \( 2^{\frac{2}{3}+\frac{3}{2}} x ^{1+\frac{1}{2}+\frac{1}{3}}=16 \)
\( \Rightarrow \) \( x=2 \) \( \Rightarrow \) \( z=2 \) si \( y=4 \)
\( z=2^{\frac{2}{3}} x^{\frac{1}{3}} \) , \( y=2^{\frac{3}{2}} x^{\frac{1}{2}} \)
Din \( x y z=16 \) si relatiile precedente obtinem: \( 2^{\frac{2}{3}+\frac{3}{2}} x ^{1+\frac{1}{2}+\frac{1}{3}}=16 \)
\( \Rightarrow \) \( x=2 \) \( \Rightarrow \) \( z=2 \) si \( y=4 \)
Politehnica University of Bucharest
The Faculty of Automatic Control and Computers
The Faculty of Automatic Control and Computers