Question

City Bus Corporation runs two buses from terminus A to terminus B, one from each terminus such that each bus makes 5 round trips in a day. There are no stops in between. These buses ply back and forth on the same route at different but uniform speeds. Each morning the buses start at 7 AM from the respective terminuses. They meet for the first time at a distance of 7 km from terminus A. Their next meeting is at a distance of 4 km from terminus B, while travelling in opposite directions. Assuming that the time taken by the buses at the terminuses is negligibly small, and the cost of running a bus is 20 per km, find the daily cost of running the buses.

• A. 3200
• B. 6800
• C. 4000
• D. 6400
• E. None of the above

Hint:

- For shuttling objects with instantaneous turnarounds, replace turns by \textit{straight continuation} on an unfolded (reflected) line; meeting points then occur at odd multiples of the first-meeting position and are reflected into the segment.
- After finding the route length, multiply by the total number of one-way legs to get total distance.

Answer 1

The Correct Option is B

Step 1: Use meeting points to get the route length.
Let \(L\) be the distance \(AB\). Let the bus from \(A\) have speed \(v_1\) and from \(B\) have speed \(v_2\). 
At the {first} meeting (moving towards each other), the position from \(A\) is \[ x_1=\frac{v_1}{v_1+v_2}L=7 \quad\Rightarrow\quad x_1=7. \] On the unfolded-line model, successive opposite-direction meetings occur at positions \(x=(2k+1)x_1\) from \(A\) (for \(k=0,1,2,\ldots\)), reflected back into \([0,L]\). 
Thus the {next} opposite-direction meeting corresponds to \(x=3x_1=21\) km from \(A\) on the unfolded line. Since this must lie between \(L\) and \(2L\) 

Step 2: Total daily distance and cost. 
Each bus makes \(5\) round trips \(\Rightarrow\) distance per bus \(=5\times 2L=10L=170\) km. 
Two buses \(\Rightarrow\) total distance \(=2\times 170=340\) km. 
Cost @ ₹\(20\) per km \(=340\times 20=\boxed{6800}\).