Heya ❤ A thief runs with a uniform speed of 100m/min.After one min. A policeman runs after a thief to catch him.He goes with a speed of 10m/min. In the first minute and increases his speed by 10m/minute every succeding minute.After how many minutes the policemen will catch thief.Kindly solve :)
Answers
≡QUESTION≡
A thief runs with a uniform speed of 10 m/minute . After one minute a policeman runs after the thief to catch him. He goes with a speed of 100 m/minute in the first minute and increases his speed by 10 m/min every succeeding minute. After how many minutes the police man will catch the thief ?
║⊕ANSWER⊕║
Let us assume total time be n minutes
Also, total distance covered by the thief in n minutes
=Speed × Time
= 100 × n
= 100n m
Distance travelled by the thief in (n+1) min = 100(n+1)m
Speed of police in 1st min= 100 m/min
Speed of police in 2nd min = 110 m/min
3rd min = 120m/min.........so on
100,110,120,............ this forms an AP
And, Total distance covered by policeman = 1st min. + 2nd min + …… (n – 1) terms
Here, we have:
a = 100
d = 110 – 100
= 10
‘n’ = n – 1
⇒200n = (n-1)[200 + (n-2)10]
⇒200n = (n – 1) [200 + 10n – 20]
⇒10n2 + 180n – 10n – 180 – 200n = 0
⇒10n2 – 30n – 180 = 0
⇒10 (n2 – 3n – 18) = 0
⇒n2 – 3n – 18 = 0
⇒n (n – 6) + 3 (n – 6) =0
⇒(n + 3) (n – 6) = 0
∴ n + 3 = 0
n = - 3
Or n – 6 = 0
n = 6
As the value of n i.e., time cannot be negative therefore n = 6
∴ Time taken by the police to catch the thief = n – 1
= 6 – 1
= 5 minutes
Answer:
Let the police catch the thief in n min
As the thief ran 1 min before the police...time taken by the thief before being caught = (n+1) min
Distance travelled by the thief in (n+1) min = 100(n+1)m
Speed of police in 1st min=100m/min
Speed of police in 2nd min=110m/min
...3rd min = 120m/min.........so on
100,110,120,............ this forms an AP
Total distance travelled by the police in n min = n/2(2 x 100 +(n-1)10)
On catching the thief by police,distance traveled by thief= distance travelled by the police
100(n+1)= n/2(2 x 100 + (n-1)10)
100n + 100= 100n +n/2(n-1)10
100=n(n-1)5
n2 -n-20 = 0
(n-5)(n+4) = 0
n-5 = 0
n= 5 OR n= -4...(but this is not possible)
so, n= 5
Time taken by the policeman to catch the thief = 5min