Hello, MathLearner!

Aruneem travels from Patna to Calcutta, a distance of 200 km at the speed of 40 km/hr.

At the same time, Nilanjan starts from Calcutta at a speed of 20 km/hr along a road

which is perpendicular to the road on which Aruneem is travelling.

When will Aruneem and Nilanjan be closest to each other?

Answer: in 4 hours Code:

40t A 200-40t
P * - - - - - * - - - - - - - - * C
* |
* |
* | 20t
* |
* |
* B

Aruneem starts at $\displaystyle P$ and travels toward $\displaystyle C$ at 40 km/hr.

. . After $\displaystyle t$ hours, he has travelled $\displaystyle 40t$ km to point $\displaystyle A.$

Note that: .$\displaystyle AC \:=\:200-40t$

At the same time, Nilanjan starts at $\displaystyle C$ and travels south at 20 km/hr.

. . After $\displaystyle t$ hours, he has travelled $\displaystyle 20t$ km to point $\displaystyle B.$

Their distance is $\displaystyle AB$, the hypotenuse of right triangle $\displaystyle ACB.$

Hence: .$\displaystyle D \;=\;AB^2 \:=\:(200-40t)^2 + (20t)^2 \:=\:2000t^2 - 16,000t + 40,000$

Then: .$\displaystyle D' \;=\;4000t - 16,000 \:=\:0 \quad\Rightarrow\quad t \,=\,4$

Therefore, minimum distance occurs in 4 hours.