Hello, Climberboy123!

At noon, ship $\displaystyle a$ is 100 km west of ship $\displaystyle b$.

Ship $\displaystyle a$ is sailing south at 35 km/h and ship $\displaystyle b$ is sailing north at 5 km/h.

How fast is the distance between the ships changing at 6:00 PM? Code:

* B
* |
* | 5t
* |
P * - - - - - * - - - * Q
| * :
| * :
35t | * : 35t
| * :
| * :
A * - - - - - - - - - * R
100

Ship $\displaystyle a$ starts at $\displaystyle P$ and sails south at 35 km/hr,

. . In $\displaystyle t$ hours, it has sailed $\displaystyle 35t$ km to point $\displaystyle A.$

Ship $\displaystyle b$ starts at $\displaystyle Q$ and sails north at 5 km/hr.

. . In $\displaystyle t$ hours, its has sailed $\displaystyle 5t$ km to point $\displaystyle B.$

Note that: .$\displaystyle PQ = AR = 100$ km.

Let: $\displaystyle x \,=\,AB.$

In right triangle $\displaystyle BRA\!:\;\;x^2 \;=\;(40t)^2 + 100^2 \quad\Rightarrow\quad x \;=\;(1600t^2 + 10,\!000)^{\frac{1}{2}} $

Can you finish it now?