Ship A is moving east at 20 miles per hour, while ship B is moving north at 15 miles per hour. At noon ship A was 5 miles east of an island, and ship B was 75 miles south of the island.

Question

At what rate is the distance of the ships changing at 1 pm?

Answer 1

x = 5 + 20 t
y = -75 + 15 t
z =distance =sqrt(25+5625)
(I bet you meant 7.5 miles but oh well)

dx/dt = 20
dy/dt = 15

at t = 0 x = 5 and y = -75
z^2 = x^2 + y^2
2 z dz = 2x dx + 2 y dy
so
dz = (x dx + y dy)/z
and dz/dt = (x dx/dt + ydy/dt)/z
= (5*20-75*15)/sqrt(25+56250