Question

Multimile {a;b;c} si {2;4;5} sunt invers proportionale, iar produsul a•b•c = 25.Calculati a,b,c.​

Answer 1

Răspuns:

a = ∛5 ; b = 2∛5 ; c = 5∛5/2

Explicație pas cu pas:

a/2 = b/4 = c/5

a·b·c = 25

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b = 4a/2 = 2a

c = 5a/2

a·2a·5a/2 = 25 =>

5a³ = 25 => a³ = 5 => a = ∛5

b = 2∛5 ; c = 5∛5/2

a·b·c =  ∛5·2∛5·5∛5/2 = ∛125·5 = 5·5 = 25

Answer 2

$\it \{a,\ b,\ c\}\ i.\ p.\ \{2,\ 4,\ 5\} \Rightarrow\ 2a=4b=5c \Rightarrow\ \begin{cases} \it 2a=4b\bigg|_{:2} \Rightarrow a=2b\\ \\ \it5c=4b \Rightarrow\ c=\dfrac{4}{5}b\end{cases}\\ \\ \\ a\cdot b\cdot c=25 \Rightarrow\ 2b\cdot b\cdot\dfrac{4}{5}b=25 \Rightarrow\ \dfrac{8}{5}b^3=25 \Rightarrow\ b^3=25\cdot\dfrac{5}{8} \Rightarrow\ \\ \\ \\ \Rightarrow\ b^3=\dfrac{125}{8} \Rightarrow\ b^3=\dfrac{5^3}{2^3} \Rightarrow\ b^3=\bigg(\dfrac{5}{2}\bigg)^3 \Rightarrow\ b=\dfrac{5}{2}$

$\it a=2b=2\cdot\dfrac{5}{2}=5;\ \ \ c=\dfrac{4}{5}\cdot b=\dfrac{4}{5}\cdot\dfrac{5}{2}=\dfrac{20}{10}=2$

$\it Numerele \ cerute\ sunt:\ \ a=5,\ \ b=\dfrac{5}{2},\ \ c=2\ .$