Question

1. Din doua puncte, situate la o distanta de 9 km unul de celalalt, pornesc in acelasi timp doi biciclisti. Daca merg unul catre altul, se intalnesc dupa 20 minute de la plecare; daca merg in acelasi sens, al doilea il ajunge pe primul dupa 3 ore. Cu ce viteza merge fiecare din acesti biciclisti?

2. La o cursa de automobile, unul din cele 3 automobile care au plecat in acelasi timp, facand pe ora cu 15 km mai putin decat primul si cu 3 km pe ora mai mult decat al treilea, a sosit la punctul de sosire cu 12 minute mai tarziu decat primul si cu 3 minute inaintea celui de-al treilea; pe drum n-au fost opriri.
Sa se afle:
a) pe ce distanta s-a facut cursa;
b) viteza fiecarui automobil;
c)durata alergarii fiecarui automobil.

Answer 1

1. $V= \frac{ \Delta d}{\Delta t}$
merg unul catre altul, se intalnesc dupa 20 minute de la plecare=> 
t1=t2=20 min
 $V_{1} = \frac{d_1}{t_1} = \frac{d_1}{20}~=\ \textgreater \ d_1=20V_1$ 
$V_{2} = \frac{d_2}{t_2} = \frac{d_2}{20} = \frac{9-d_1}{20} =\ \textgreater \ d_1=9-20V_2$
=>$20V_1=9-20V_2$      (1)

merg in acelasi sens, al doilea il ajunge pe primul dupa 3 ore=> t3=180min
$V_1= \frac{d_3}{180} =\ \textgreater \ d_3=180V_1$
$V_2= \frac{9+d_3}{180}=\ \textgreater \ d_3=180V_2-9$
=> $180V_1=180V_2-9$   (2)

(1) imultita cu 9 ⇒$180V_1=81-180V_2$   (3)
  
si avem (3) +(2) => $360V_1=91-9=72 =\ \textgreater \ V1=0,2~Km/min=\ \textgreater$
$\boxed {V_1=12~Km/ora}$
$20V_1=0,2*20=9-20V_2=\ \textgreater \ 20V_2=9-4=5$
$V_2=0,25~Km/min$
$\boxed{V_2=15~Km/ora}$

2. $\left \{ {{V_2=V_1-15~~=\ \textgreater \ V_1=V_2+15} \atop {V_2=V_3+3~~=\ \textgreater \ V_3=V_2-3}} \right.$
transformam: 
12 min=0,2 ore
3 mi=0,05 ore

$\left \{ {{t_2=t_1+0.2~~=\ \textgreater \ t_1=t_2-0.2} \atop {t_2=t_3-0.05~~=\ \textgreater \ t_3=t_2+0.05}} \right.$

 $V= \frac{ \Delta d}{\Delta t}$
distata este aceeasi :d
$V_2= \frac{d}{t_2}$
$V_1= \frac{d}{t_1} ~~=\ \textgreater \ V_2+15= \frac{d}{t_2-0.2}$
$V_3= \frac{d}{t_3} ~~=\ \textgreater \ V_2-3= \frac{d}{t_2+0.05}$

=> $d=V_2t_2$
$d=(V_2+15)(t_2-0.2)=V_2t_2+15t_2-0.2V_2-3$
$d=(V_2-3)(t_2+0.05)=V_2t_2-3t_2+0.05V_2-0.15$

=>$\left \{ {{V_2t_2=V_2t_2+15t_2-0.2V_2-3~~=\ \textgreater \ 15t_2-0.2V_2=3} \atop {V_2t_2=V_2t_2-3t_2+0.05V_2-0.15~~=\ \textgreater \ -3t_2+0.05V_2=0.15~~~~|*5}} \right.$

$\left \{ {{15t_2-0.2V_2=3} \atop {-15t_2+0.25V_2=0.75}} \right.$
=> $0.05V_2=3.75=\ \textgreater \ \boxed{V_2=75~Km/ora}$
$V_1=75+15=\ \textgreater \ \boxed{V_1=90~Km/ora}$
$V_3=75-3=\ \textgreater \ \boxed{V_3=72~Km/ora}$

$15t_2-0.2-75=3~=\ \textgreater \ t_2=1,2 ore~=\ \textgreater \ \boxed{t_2=72~min}$
$t_1=72-12~=\ \textgreater \ \boxed{t_1=60~min}$
$t_3=72+3~=\ \textgreater \ \boxed{t_3=75~min}$

$d=V_2t_2=75*1.2=\ \textgreater \ \boxed{d=90~Km}$