Question

22. A theif runs with a uniform speed of 100 m/minutes. After one minute a policeman
runs after the thief to catch him. He goes with a speed of 10 m/minute in the first
minute and increases his speed by 10 m/minute every succeeding minute. After
how
many minutes the policemen will catch the thief?​

Answer 1

Answer:

$\bigstar{\bold{Time\:taken=5\:minutes}}$

Step-by-step explanation:

$\Large{\underline{\underline{\bf{Given:}}}}$

• Speed of thief = 100 m/min
• Speed of policeman = 10 m/min in the first minute and increases by 10m/min in every succeeding minutes.

$\Large{\underline{\underline{\bf{To\:Find:}}}}$

• Minutes taken for policeman to catch the thief

$\Large{\underline{\underline{\bf{Solution:}}}}$

→ Let the policeman man take t minutes to catch the thief.

→ Given the policeman runs after 1 min to catch the thief,

  Time taken by the thief before being caught = ( t + 1) min

→ Distance travelled by the thief is given by,

  Distance = Speed × Time

→ Substitute the given datas,

  Distance travelled by thief = 100 × ( t + 1 )m ----(1)

→ The speed of the policemen in each second is given by

  100m/min , 110m/min , 120 m/min....

→ This forms an A.P

→ Hence the sum of the A.P is the distance covered by the policeman.

→ Sum of n terms of an A.P is given by

  $S_n=2a_1+(n-1)\times d$

→ Substituting thhe datas,

  $S_n=\dfrac{n}{2} (2\times100+(n-1)\times 10)$

 $Distance\:travelled\:by\:policeman=\dfrac{n}{2}(200+10n-10)$

 Distance travelled by the policeman = n/2 (190 + 10 n)

                                                              = t/2 ( 190 + 10 t )

→ Distance travelled by the thief = Distance travelled by policeman

→ Substituting equation 1 and 2

  100 (t + 1) = t/2 (190 + 10 t)

  100t + 100 = t/2 (190 + 10t)

→ Multiplying by 2

  200t + 200 = t (190 + 10t)

  200t + 200 = 190t + 10t²

  200t - 190 t + 200  = 10t²

  10 t + 200 = 10 t²

→ Dividing equation by 10

  t + 20 = t²

  t² - t - 20 = 0

→ Solving by splitting the middle term

  t² - 5t + 4t - 20 = 0

  t ( t - 5 ) + 4(t - 5) = 0

  (t - 5 ) ( t + 4) = 0

→ Either

  t + 4 = 0

  t = -4

→ This can't happen since time can't be negative

→ Or

   t - 5 = 0

   t = 5

→ Hence policeman takes 5 minutes to catch the thief

$\boxed{\bold{Time\:taken=5\:minutes}}$