Question

a theif after commuting a theft runs at a uniform speed of 50 ms and after 2 mins a policeman starts running at him with a speed of 60ms with an increasing speed of 5ms .find time taken by the police to catch the theif
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Answer 1

Answer:

According to the question,

Distance between thief and police after 2 minutes = 100 m

Police starts chasing thief with initial speed of 60 and increasing his speed by 5 m every minute.

Let the police chase the thief for t minutes.

Implies, thief covers total 100 + 50t m

Police runs with following pattern 60,65,70...

This is an Arithmetic Progression with a = 60, d = 5 & number of terms = t

t/2(2×60+(t−1)×5)=50t+100

t×(120+5t−5)=100t+200

5t

2

+15t−200=0

t

2

+3t−40=0

t={−3±

(9+160)}/2

=(−3±13)/2=+5

Implies, police will catch thief in 5 minutes.

or the thief is caught after 7 minutes.

Answer 2

Answer:

5 minutes

Step-by-step explanation:

the distance covered by the theif per minute forms an A.P, from after 2 minutes(2×50=100):

$50t+100,~~for~time=t$

the police ran after 2 minutes with initial velocity of 60m/min and and acceleration  of $5ms^{-2}$. It also forms an ap:

$60,65,70,....,60+(t-1)5$

$Where,~a=60~and~d=5$

and total distance at time t:

$S_t=\frac{t}{2}[2(60)+(t-1)5]$

equating both=

$50t+100=\frac{t}{2} [120+(t-1)5]\\\rightarrow 100t+200=120t+(t-1)5t\\\rightarrow 100t+200=120t+5t^2 -5t\\\rightarrow 0=15t+5t^2-200\\\rightarrow 0=5(t^2+3t-40)\\\rightarrow t^2+3t-40=0\\\rightarrow t^2+8t-5t-40=0\\\rightarrow t(t+8)-5(t+8)=0\\\rightarrow (t-5)(t+8)=0\\\rightarrow t=5~or~t=-8$

as time can't be negative, we take $t=5min$

$\therefore$ police will catch the theif after 5min