Question

A thief, after committing a theft, runs at a uniform speed of 50 m/minute. After 2 minutes, a policeman runs to catch him. He goes 60 m in first minute and increases his speed by 5 m/minute every succeeding minute. After how many minutes, the policeman will catch the thief?

Answer 1

Suppose the policeman catches the thief after t minutes.

Uniform speed of the thief = 50 m/min

∴ Distance covered by thief in (t + 2) minutes = 50 m/min × (t + 2) min = 50 (t + 2) m

The distance covered by the policeman in t minutes is in AP, with 60 and 5 as the first term and the common difference, respectively.

Now,

Distance covered by policeman in t minutes = Sum of t terms

= `t/2`[2 × 60 + (t − 1) × 5]

= `t/2`[115 + 5t] m

When the policeman catches the thief, we have

`t/2`[115 + 5t] = 50(t + 2)

115t + 5t² = 100t + 200

⇒ 5t² + 15t − 200 = 0

⇒ t² + 3t − 40 = 0

⇒ (t + 8)(t − 5) = 0

So, t = −8 or t = 5

∴ t = 5  (As t cannot be negative)

Thus, the policeman catches the thief after 5 min.