# Tourist Jirka

Distance between the points A and B is 13.5 km. Jirka went from point A to point B unknown speed and for an unknown period of time. Back to the point A went slower by 3 km/h which means that went 20 minutes more.

How long Jirka took the return journey?

Correct result:

t2 =  1.4 h

#### Solution:

$13.5 = v_1 t_1 \ \\ 13.5 = v_2t_2 = (v_1-3)(t_1+20/60) \ \\ \ \\ 3x^2 +x -4.5 =0 \ \\ \ \\ a=3; b=1; c=-4.5 \ \\ D = b^2 - 4ac = 1^2 - 4\cdot 3 \cdot (-4.5) = 55 \ \\ D>0 \ \\ \ \\ x_{1,2} = \dfrac{ -b \pm \sqrt{ D } }{ 2a } = \dfrac{ -1 \pm \sqrt{ 55 } }{ 6 } \ \\ x_{1,2} = -0.16666667 \pm 1.2360330811826 \ \\ x_{1} = 1.0693664145159 \ \\ x_{2} = -1.4026997478493 \ \\ \ \\ \text{ Factored form of the equation: } \ \\ 3 (x -1.0693664145159) (x +1.4026997478493) = 0 \ \\ \ \\ t_1 = x_1 = 1.0693664145159 \ h \ \\ t_2 = t_1 + 20/60 = 1.4 \ \text{h} \ \\ v_1 = 13.5/t_1 = 12.62 \ km/h \ \\ v_2 = 13.5/t_2 = 9.62 \ km/h$

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