Question

Numerele rationale a,b si c sunt direct proportionale cu 2,4 si 5 iar 5a+3b+2c=64.Produsul numerelor a,b si c este...​

Answer 1

$\it \{a,\ b,\ c\}\ d.\ p.\ \{2,\ 4,\ 5\} \Rightarrow \dfrac{a}{2}=\dfrac{b}{4}=\dfrac{c}{5}=k \Rightarrow \begin{cases} \it a=2k\\ \\ b=4k\\ \\ c=5k\end{cases}\ \ \ \ \ \ (*)$

$\it 5a+3b+2c=64\ \stackrel{(*)}{\Longrightarrow}\ 5\cdot 2k+3\cdot 4k+2\cdot 5k=64 \Rightarrow 10k+12k+10k=64 \Rightarrow\\ \\ \Rightarrow32k=64|_{:32} \Rightarrow k=2\ \stackrel{(*)}{\Longrightarrow}\ \begin{cases} \it a=2\cdot2=4\\ \\ \it b=4\cdot2=8\\ \\ \it c=5\cdot2=10\end{cases}\\ \\ \\ a\cdot b\cdot c=4\cdot8\cdot10=320$