Question

At noon, ship A is 20 kilometers north of Ship B. Ship A is traveling south at 50 kilometers per hour, and ship B is traveling east at 40 kilometers per hour. If visibility is 10 kilometers, could the people on the two ships ever see each other?

Answer 1

The question breaks down into finding the shortest distance between the two ships.

make a sketch of the current position, letting O be the current position of ship B
after t hours, Ship A will be (20-50t) km north of O and ship B will be 40t km east of O
This forms a right-angled triangle, and
AB^2 = (20-50t)^2 + (40t)^2
2 AB d(AB)/dt = 2(20-50t)(-50) + 2(40t)(40) , divide each term by 2 and simplify ...
d(AB)/dt = (-1000 + 2500t + 1600t) / AB = 0 for a min of AB

4100t = 1000
t = .2439

AB^2 = (20-50(.2439))^2 + (40(.2439))^2
AB = 12.49 km

Since visibility is only 10 km, ........