Hello, racuna!
I had to baby-talk through it . . . Is anyone surprised?
A boat leaves a dock at 2:00p.m. and travels due south at a speed of 20kph.
Another boat has been heading due east at 15 kph and reaches the same dock at 3:00p.m.
At what time were the two boats closest together?
Here's my reasoning . . .
At 2:00 the first boat
#1 is at the dock
P.
. . It moves south at 20 kph.
. . In
t hours,
#1 will be
20t km south of the dock at point
A.
The second boat
#2 had been moving east at 15 kph since 2:00 and reaches the dock at 3:00.
. . Back at 2:00,
#2 must have been 15 km west of the dock at point
Q.
. . In
t hours,
#2 will be
15t km
closer to the dock at point
B.
Code:
. . .| - - - 15 - - - |
Q 15t B 15-15t P
*- - - - * - - - - *
\ |
\ |20t
d \ |
\ |
*A
The distance between the boats is:
d=AB
From Pythagorus, we get: \(\displaystyle \;d^2\:=\

20t)^2\,+\,(15-15t)^2\)
. . which simplifies to: \(\displaystyle \;d\:=\

625t^2\,-\,450t\,+\,225)^{\frac{1}{2}}\)
And
that is the function we must minimize.