Question

Determinați numerele raționale pozitive a, b şi c în fiecare din cazurile: a) (a, b, c) d. P. (2, 3, 5), iar abc = 3,75; b) (a, b, c) d. P. (1, 0,5, 0,(3)), iar ab + bc + ca = 1. Vă rog dau coroană. ​.

Answer 1

Explicație pas cu pas:

a)

$\dfrac{a}{2} = \dfrac{b}{3} = \dfrac{c}{5} = k$

$a = 2k \ ; \ b = 3k \ ; \ c = 5k$

$abc = 2k \cdot 3k \cdot 5k = 30k^{3} \\ 30k^{3} = 3.75 \iff k^{3} = \dfrac{1}{8} \implies k = \dfrac{1}{2}$

$\implies a = 1 \ ; \ \ b = \dfrac{3}{2} \ ; \ c = \dfrac{5}{2}$

b)

$0,5 = \dfrac{5}{10} = \dfrac{1}{2} \\ 0,(3) = \dfrac{3}{9} = \dfrac{1}{3}$

$\dfrac{a}{1} = \dfrac{b}{\dfrac{1}{2}} = \dfrac{c}{\dfrac{1}{3}} = k$

$a = k \ ; \ b = \dfrac{k}{2} \ ; \ c = \dfrac{k}{3}$

$ab + bc + ca = k \cdot \dfrac{k}{2} + \dfrac{k}{2} \cdot \dfrac{k}{3} + \dfrac{k}{3} \cdot k = \dfrac{3 {k}^{2} + {k}^{2} + 2 {k}^{2}}{6} = \dfrac{6 {k}^{2}}{6} = {k}^{2}$

${k}^{2} = 1 \implies k = 1$

$\implies a = 1 \ ; \ b = \dfrac{1}{2} \ ; \ c = \dfrac{1}{3}$