The Fellowship of the Ring was broken when Boromir momentarily
betrayed Frodo Baggins. By the time Aragorn tried to find Frodo, he and Sam were
already 1 km to the east crossing the lake. Since Frodo’s destiny was out of his
hands, Aragorn took the remaining members to hunt down the Orcs. If Aragorn’s
company was moving north at 12 km/h while Frodo and Sam were moving east 4
km/h, how far was the distance between the two groups changing 1 hour later?
At t=0, let's say that
Frodo was at (1,0), 1 km east of
Aragorn, who was at (0,0)
At time t hours later, the two parties were at
(1+4t,0) and (0,-12t)
So, the distance d(t), between the parties is given by
d^2 = (1+4t)^2 + (-12t)^2
d^2 = 160t^2 + 8t + 1
At t=1, d=13
Now, the rate of change of d is given by
2d dd/dt = 320t+8
Now just plug in the numbers
2*13 dd/dt = 328
dd/dt = 328/26 = 12.6 km/hr
This makes sense, since the faster speed of 12 km/hr contributes much more to the distance than the much slower speed of 4 km/hr.