Notam cu:
L= lungimea initial
a a dreptunghiului
l = latimea initial
a a dreptunghiului
Aria dreptunghiului = L · l
Perimetrul dreptunghiului = 2·( L + l)
[tex]\bf (L+3)\cdot (l+2) = L\cdot l + 47[/tex] (relatia 1)
[tex]\bf (L-3)\cdot (l+3) = L\cdot l[/tex] (relatia 2)
calculam in prima relatie
[tex]\bf L\cdot l+2L+3l+6 = L\cdot l + 47\\\\2L+3l+6=47\\\\2L+3l=47-6\\\\2L+3l = 41[/tex]
calculam in a doua relatie
[tex]\bf (L-3)\cdot (l+3) = L\cdot l\\\\ L\cdot l+3L-3l-9= L\cdot l\\\\3L-3l-9 = 0\\\\3L-3l=9\:\:|:3\\\\L-l=3\\\\L=l+3[/tex]
Am aflat ca: L = l +3 si 2L +3l = 41 vom inlocui pe L din prima relatie in a doua relatie si vom afla latimea
[tex]\bf 2 \cdot (l+3)+3l=41\\\\2l+6+3l=41\\\\5l+6=41\\\\5l=41-6\\\\5l=35\:\:|:5[/tex]
[tex]\boxed{\bf l= 7\:m}[/tex]
[tex]\bf L = 7+3[/tex]
[tex]\boxed{\bf L = 10\:m}[/tex]
[tex]\bf Aria = 7\cdot 10[/tex]
[tex]\boxed{\bf Aria = 70\:m^{2}}[/tex]
[tex]\bf Perimetrul = 2\cdot(7+10)[/tex]
[tex]\bf Perimetrul = 2\cdot 17[/tex]
[tex]\boxed{\bf Perimetrul = 34\:m}[/tex]
Verificare:
(10 + 3) · (7 + 2) = 70 + 47
13 · 9 = 70 + 47
117 = 117 (adevarat)
