A train 7

A train overtakes two persons walking in the same direction as the train at speeds of 2 km/h and 4 km/h. The train passes each of them completely in 9 and 10 seconds respectively. Find the length of the train.

Final Answer:

x =  50 m

Step-by-step explanation:

$$\begin{gathered} v_1=2\text{km/h}\to \text{m/s}=2:3.6\text{m/s}=0.55556\text{m/s} \\ {v}_2=4\text{km/h}\to \text{m/s}=4:3.6\text{m/s}=1.11111\text{m/s} \\ {t}_1=9\text{s} \\ {t}_2=10\text{s} \\ {t}_1=\frac{x}{v-v_1} \\ {t}_2=\frac{x}{v-v_2} \\ {t}_1\cdot (v-v_1)=x \\ {t}_2\cdot (v-v_2)=x \\ 9\cdot (v-0.55555555555556)=x \\ 10\cdot (v-1.1111111111111)=x \\ 9v-x=5 \\ 10v-x=11.111111 \\ Pivot:Row1\leftrightarrow Row2 \\ 10v-x=11.11 \\ Row2-\frac{9}{10}\cdot Row1\to Row2 \\ -0.1x=-5 \\ x=\frac{-5}{-0.1}=50 \\ v=\frac{11.11111111+x}{10}=\frac{11.11111111+50}{10}=6.11111111 \\ v=\frac{55}{9}\doteq 6.111111 \\ x=50=50\text{m} \end{gathered}$$

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