Question

Optimization

At 1:00 PM ship A is 30 miles due south of ship B and is sailing north at a rate of 15mph. If ship B is sailing due west at a rate of 10mph, at what time will the distance between the two ships be minimal? will the come within 18 miles of each other?


The answer is 2:23:05 and yes,
i just need help with the setup

Answer 1

The positions of A and B are parametric functions of time.

Take the position of ship A at 1:00 pm be (0,0), the
xa(t) = 0
ya(t) = 10t

xb(t) = -15t
yb(t) = 30

Distance between the two ships
D(t) = √((xb(t)-xa(t))²+(yb(t)-ya(t))²)

Substitute xa,xb,ya,yb in the expression for D(t) and differentiate with respect to t.

For the minimum distance, D'(t0) = 0.
Verify that D(t0) is a minimum by confirming that D"(t0)>0, or by comparing values of D'(t0-) and D'(t0+) with D'(t0).

Evaluate D(t0) for the remainder of the question.