Hello, imthatgirl!

2) A train leaves the station at 10:00 and travels due north at 100km/h.

Another train has been heading due west at 120km/h and reaches the station at 11:00.

At what time were the 2 trains closest?

Train #1 leaving the station $\displaystyle S$ at 100 km/hr.

In $\displaystyle t$ hours, it is at $\displaystyle A\!:\;SA = 100t$

Train #2 heads west at 120 km/hr and reaches $\displaystyle S$ at 11:00.

. . Hence, at 10:00, it was at $\displaystyle P$, 120 km east of $\displaystyle S.$

In $\displaystyle t$ hours, it is at $\displaystyle B\!:\;PB = 120t\quad\Rightarrow\quad SB = 120-120t$ Code:

A *
| *
| *
100t | *
| *
| *
* - - - - - * - - - - - *
S 120-120t B 120t P
: - - - - 120 - - - - :

The distance is: . $\displaystyle d \;=\;AB^2\;=\;(100t)^2 + (120-120t)^2$

Hence: .$\displaystyle d \;=\;24,400t^2 - 28,800t + 14,400$

Then: .$\displaystyle d\,' \;=\;48,800t - 28,800 \;=\;0 \quad\Rightarrow\quad t \:=\:\frac{36}{61}\text{ hour} \;\approx\;35\text{ minutes}$

Therefore, the trains were the closest at about $\displaystyle \boxed{10:35}$