Question

A thief escaped from police custody. Since he was a sprinter he could clock 40 km/hr. The police realized it after 3hr and started chasing him in the same direction at 50 km/hr. The police had a dog which would rum at 60 km/hr. The dog could run to the thief and then return to the police and then would turn back towards the thief. If kept on doing so till the police caught the thief. Find the total distance travelled by the dog in the direction of the thief
(a) 720km
(b) 600km
(c) 660km
(d) 360km

Answer 1

Hint: Let ‘t’ be the time taken by the police to catch the thief. Distance travelled by police is equal to distance travelled by the thief. After getting the value of t. Find the distance travelled by police and dog. The sum of distance travelled by police and the to and fro distance of dog is the total distance covered by the dog.

Complete step-by-step answer:
Speed of the thief = 40 km/hr.
Speed of the police = 50 km/hr.
Speed of the dog = 60 km/hr.
Let ‘t’ be the time taken for the police to catch up with the thief. Let us take the speed of the thief as \[\dfrac{1}{t}\] and speed of police as \[{{V}_{p}}\].
We have been told that the police started 3hrs after the thief started running. After 3 hours, the thief is still in spring.
Distance travelled by police = Distance travelled by thief + distance travelled by thief in 3hrs. – (1)
We know that Distance = Speed \[\times \] time.
Distance travelled by police = Speed of police \[\times \] time = 50t.
Distance travelled by thief = Speed of thief \[\times \] time = 40t.
Distance travelled by thief in 3hrs = Speed of thief \[\times \] time = 40 \[\times \] 3 = 120.
Now let us substitute all this in (1).
\[\begin{align}
  & 50t=40t+120 \\
 & \Rightarrow \left( 50-40 \right)t=120 \\
\end{align}\]
\[t=\dfrac{120}{10}=12\]hrs.
\[\therefore \] The time taken by police to catch the thief is 12hrs.
Distance travelled by dog = Speed of dog \[\times \] time taken = 60 \[\times \] 120 = 720km.
720km is the distance travelled by police and distance travelled by the dog to and from the police.
Distance travelled by the police = 50 \[\times \] 12 = 600km.
\[\therefore \] Distance travelled by police + 2 \[\times \] distance by dog = 720km.
600 + 2 \[\times \] distance by dog = 720.
Distance by dog = \[\dfrac{720-600}{2}=\dfrac{120}{2}=60\].
Total distance towards thief = distance travelled by police + distance by dog
\[\therefore \] Total distance towards thief = 600 + 60 = 660km.
\[\therefore \] Total distance travelled by dog in direction of thief is 660km.
\[\therefore \] Option (c) is the correct answer.

Note: To to and fro distance covered by the dog is equal. The “Fro” distance is the distance from thief to police and “To” distance is the distance from police to thief after once it returns. Thus both the distances are equal.