Two trains 'A' and 'B' of length '$l$' and '$4 l$' are travelling into a tunnel of length '$\mathrm{L}$' in parallel tracks from opposite directions with velocities $108 \mathrm{~km} / \mathrm{h}$ and $72 \mathrm{~km} / \mathrm{h}$, respectively. If train 'A' takes $35 \mathrm{~s}$ less time than train 'B' to cross the tunnel then. length '$L$' of tunnel is :

(Given $\mathrm{L}=60 l$ )

Let's start by converting the velocities of both trains to m/s:<br/><br/> Train A: $$108 \frac{km}{h} \times \frac{1000 m}{km} \times \frac{1 h}{3600 s} = 30 \frac{m}{s}$$<br/><br/> Train B: $$72 \frac{km}{h} \times \frac{1000 m}{km} \times \frac{1 h}{3600 s} = 20 \frac{m}{s}$$ <br/><br/> To cross the tunnel, Train A has to cover a distance equal to the length of the tunnel plus its own length: $L + l$<br/><br/> Similarly, Train B has to cover a distance equal to the length of the tunnel plus its own length: $L + 4l$ <br/><br/> We are given that Train A takes 35 seconds less time than Train B to cross the tunnel. Let's denote the time taken by Train A as $t_A$ and the time taken by Train B as $t_B$. Then, we have: <br/><br/> $t_B = t_A + 35$ <br/><br/> Using the formula distance = velocity × time, we can write the equations for both trains: <br/><br/> Train A: $(L + l) = 30t_A$<br/><br/> Train B: $(L + 4l) = 20t_B$ <br/><br/> Now, we can substitute $t_B = t_A + 35$ in the equation for Train B: <br/><br/> $(L + 4l) = 20(t_A + 35)$ <br/><br/> We have two equations and two unknowns ($L$ and $t_A$). We can solve this system of equations by eliminating one of the unknowns. Let's eliminate $t_A$ by expressing it from the equation for Train A: <br/><br/> $t_A = \frac{L + l}{30}$ <br/><br/> Now, substitute this expression for $t_A$ in the equation for Train B: <br/><br/> $(L + 4l) = 20\left(\frac{L + l}{30} + 35\right)$ <br/><br/> Multiplying both sides by 30 to get rid of the fraction: <br/><br/> $30(L + 4l) = 20(L + l) + 20 \times 35 \times 30$ <br/><br/> Expanding the equation: <br/><br/> $30L + 120l = 20L + 20l + 21,000$ <br/><br/> Simplify: <br/><br/> $10L + 100l = 21,000$ <br/><br/> Since we are given that $L = 60l$, substitute this into the equation: <br/><br/> $10(60l) + 100l = 21,000$ <br/><br/> Solve for $l$: <br/><br/> $700l = 21,000$ <br/><br/> $l = 30$ <br/><br/> Now, substitute the value of $l$ back into the equation for $L$: <br/><br/> $L = 60l = 60 \times 30 = 1800$ <br/><br/> So, the length of the tunnel is: <br/><br/> 1800 m