SOLUTION: A ship is travelling north at a constant speed of 40 km/h. A second ship, which is initially 60 km to the east of the first ship, is travelling northwest at a speed of 50 km/h. Wha

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Question 1192807: A ship is travelling north at a constant speed of 40 km/h. A second ship, which is initially 60 km to the east of the first ship, is travelling northwest at a speed of 50 km/h. What is the minimum distance between these two ships?
Found 2 solutions by greenestamps, math_tutor2020:
Answer by greenestamps(13195) About Me  (Show Source):
You can put this solution on YOUR website!


Let the starting position of the first ship be (0,0).

Then from the given information, the position of the first ship after t hours is (0,40t).

The initial position of the second ship is (60,0); it is moving 50 km/h in the northwest direction, which is -25sqrt%282%29 km/h in the x direction and 25sqrt%282%29 km/h in the y direction.

So the position of the second ship after t hours is (60-%2825sqrt%282%29%29t,25sqrt%282%29t).

The distance between the two ships after t hours is then the distance between the points (0,40t) and (60-%2825sqrt%282%29%29t,25sqrt%282%29t). That distance, by the distance formula (AKA Pythagorean Theorem) is

sqrt%28%2860-%2825sqrt%282%29%29t%29%5E2%2B%2840t-25sqrt%282%29t%29%5E2%29

You could find the minimum distance between the two ships by finding where the derivative of that distance function is zero; but if is far easier to find the minimum value by graphing the function on a graphing calculator.

The answer I got using my TI-83 calculator was a minimum distance between the ships of about 3.65 km at t=1.69 hours.

ANSWER: about 3.65km

Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

I agree with the set up @greenestamps has, but I'm getting a slightly different answer.

The minimum of

occurs at the approximate point (t,f(t)) = (1.6683, 7.8151)

In other words, the minimum distance of roughly 7.8151 km occurs at about the t = 1.6683 hour mark.