what is the answer to this?

At noon, ship A is 40 nautical miles due west of ship B. Ship A is sailing west at 18 knots and ship B is sailing north at 25 knots. How fast (in knots) is the distance between the ships changing at 6 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)

Note: Draw yourself a diagram which shows where the ships are at noon and where they are “some time” later on. You will need to use geometry to work out a formula which tells you how far apart the ships are at time t, and you will need to use “distance = velocity * time” to work out how far the ships have travelled after time t.

30.43812587 knots/hour or to the proper 2 significant digits: 30 knots/hour

First visualize the problem: Ship A is to the west of Ship B and sailing further west so its distance from a the starting point of Ship B is the 40 nautical miles plus 18 knots times the number of hours sailed until you measure the rate of change. Ship B is easier as it is simply 25 knots times the number of hours sailed. Call that number of hours “t” and realize they’re paths form a right triangle with the paths as the legs making the right angle and the hypotenuse being the distance between them.

So, the measure of the hypotenuse at the chosen time is the distance. The later portion of the Question gives noon as the start time for the problem and we know their rate at 6 pm is desired. So we have:

hypotenuse^2 = h^2 = (18*t + 40)^2 + (25t)^2….expand the right side
h^2 = 324t^2 + 1440t + 1600 + 625t^2….simplify the right side; take square roots of both sides
h = (949t^2 + 1440t + 1600)^0.5

That’s the distance between them. We want to know the rate at which it is changing and so we need the first derivative of that. It turns out the first derivative is:

dh = (0.5) * (1898t + 1440) / (949t^2 + 1440t + 1600)^0.5
dh = (949t + 720) / (949t^2 + 1440t + 1600)^0.5

If we now substitute 6 for t, we get:

dh = (949*6 + 720) / (949*6^2 + 1440*6 + 1600)^0.5
dh = 30.43812587

So the rate of change in the distance between them at 6 pm is 30 knots / hour.

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