Trucks (10 m long) and cars (5 m long) go on a single lane bridge. There must be a gap of at least
20 m after each truck and a gap of at least 15 m after each car. Trucks and cars travel at a speed of
36 km/h. If cars and trucks go alternately, what is the maximum number of vehicles that can use the
bridge in one hour?
Options:
(A) 1440
(B) 1200
(C) 720
(D) 600
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE BT 2017 Q.63)**
**1. Governing Concept & Question Classification**:
- **Section**: General Aptitude (GA)
- **Question Type**: **MCQ** (2 Marks)
- **Topic**: Quantitative Aptitude / Speed, Distance & Traffic Flow
**2. Detailed Step-by-Step Derivation**:
- Let us calculate the length occupied by a recurring vehicle unit:
- Cars and trucks move alternately: **... - Truck - Gap - Car - Gap - Truck - ...**
- Length of a truck = $10\text{ m}$.
- Gap required after each truck = $20\text{ m}$.
- Length of a car = $5\text{ m}$.
- Gap required after each car = $15\text{ m}$.
- One full alternating repeating cycle (one Truck + one Car) requires a total spatial distance:
$L_{\text{cycle}} = 10\text{ m (truck)} + 20\text{ m (gap)} + 5\text{ m (car)} + 15\text{ m (gap)} = 50\text{ m}$
- Number of vehicles in one repeating cycle = $2$ (1 truck + 1 car).
- Speed of the traffic stream:
$v = 36\text{ km/h} = 36 \times \frac{5}{18}\text{ m/s} = 10\text{ m/s}$
In one hour ($t = 3600\text{ s}$), the total distance traveled by the stream is:
$D_{\text{1 hour}} = 36\text{ km} = 36,000\text{ m}$
- Number of full repeating 50 m cycles passing any point on the bridge in one hour:
$N_{\text{cycles}} = \frac{36,000\text{ m}}{50\text{ m/cycle}} = 720\text{ cycles}$
- Since each cycle contains 2 vehicles:
$\text{Maximum number of vehicles} = 720 \times 2 = 1440\text{ vehicles}$
- **Option-by-Option Evaluation**:
- **Option (A) [CORRECT]**: 1440 vehicles can cross the bridge per hour ($720\text{ cycles} \times 2\text{ vehicles/cycle} = 1440$).
- **Option (B) [INCORRECT]**: 1200 assumes an average vehicle plus gap of 30 m without alternating lengths.
- **Option (C) [INCORRECT]**: 720 is the number of 2-vehicle cycles, forgetting to multiply by 2.
- **Option (D) [INCORRECT]**: 600 vehicles assumes an incorrect cycle length of 60 m.
**3. Final Answer / Verification Summary**:
- **Official GATE Answer Key**: **A**
- **Scoring Criteria**: +2.0 Marks (Negative marking: -0.67 marks).
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