Question

A theif runs away from a police station with a uniform speed of $100m$/minutes. After a minute a policemen runs behind the theif to catch him. He goes at a speed of $100m/$minutes in first minute and increases his speed by $10m$ in each succeeding minute. After how many minutes, the policemen will catch the thief?

• A. $5$
• B. $6$
• C. $10$
• D. $11$

Answer 1

Answer: (A)

$5$

Let n be number of minutes after which policeman catches thief.

Then at that instant

Distance travelled by thief = Distance travelled by police ------- (1)

Now Distance travelled by thief moving uniform with $100m/min=100(n+1)$

Since policeman initially travelled with $100m/min$ then after every 1 min he increases his speed be 10 m

Hence ,

Total distance travelled by thief in 'n' minutes

$=\frac{n}{2}[2a+(n-1\left)d\right]$ $[a=100,d=10]$

$D=\frac{n}{2}[2\times 100+(n-1\left)10\right]$

$D=\frac{n}{2}[200+10n-10]=\frac{n}{2}[10n+190]$

From (1) we get

$100(n+1)=\frac{n}{2}[10n+190]$

$200n+200=10n^2+190n$

$10n^2-10n-200=0$

$n^2-n-20=0$

$n^2-5n+4n-20=0$

$(n-5)(n+4)=0$

$n=5,-4$ but $[n>0]$

Hence after 5 min policeman catches thief.