We are given:
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The train overtakes two people walking at 8 kmph and 12 kmph
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The time taken to overtake them completely is 9 seconds and 10 seconds, respectively
We need to find the length of the train.
Step-by-step approach:
Let the speed of the train = V kmph
Let the length of the train = L meters
1. Relative speed when overtaking a person:
When objects move in the same direction, relative speed = (Train speed - Person speed)
First case (person walking at 8 kmph, time = 9 s):
Relative speed = (V - 8) kmph
Convert to m/s:
= (V - 8) × (1000 / 3600) = (V - 8) × (5/18) m/s
Distance = Speed × Time
So,
L = [(V - 8) × (5/18)] × 9
=> L = (V - 8) × (5/2) — Equation (1)
Second case (person walking at 12 kmph, time = 10 s):
Relative speed = (V - 12) kmph
In m/s = (V - 12) × (5/18)
So,
L = [(V - 12) × (5/18)] × 10
=> L = (V - 12) × (25/9) — Equation (2)
Equating both expressions for L:
From Eq (1):
L = (V - 8) × (5/2)
From Eq (2):
L = (V - 12) × (25/9)
Now equate:
(V - 8) × (5/2) = (V - 12) × (25/9)
Multiply both sides by 18 to eliminate denominators:
⇒ (V - 8) × 45 = (V - 12) × 50
⇒ 45V - 360 = 50V - 600
Bring all terms to one side:
⇒ 50V - 45V = 600 - 360
⇒ 5V = 240
⇒ V = 48 kmph
Now find the length of the train:
Use Equation (1):
L = (V - 8) × (5/2)
= (48 - 8) × (5/2)
= 40 × (5/2) = 100 meters
Final Answer:
c) 100 m
