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

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) (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 (,).

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

So the position of the second ship after t hours is (,).

The distance between the two ships after t hours is then the distance between the points (,) and (,). That distance, by the distance formula (AKA Pythagorean Theorem) is

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) (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.

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