Question

Two boats leave the port at the same time with one boat traveling north at 15 knots per hour and the other boat traveling west at 12 knots per hour. How fast is the distance between the boats changing after 2 hours?

So I draw a right triangle and label the sides as x and y and the hypotenuse as D (for distance). I'm guessing I'm going to have to use the pythagorean theorem and then take the derivative but I'm not quite sure of what to do with the 2 hours data...

Thank you!

Answer 1

The distance is changing at `sqrt(1476)/2` knots per hour.

Explanation:

Let the distance between the two boats be `d` and the number of hours they've been travelling be `h`.

By the pythagorean theorem, we have:

`(15h)^2 + (12h)^2 = d^2`

`225h^2 + 144h^2 = d^2`

`369h^2 = d^2`

We now differentiate this with respect to time.

`738h = 2d((dd)/dt)`

The next step is finding how far apart the two boats are after two hours. In two hours, the northbound boat will have done 30 knots and the westbound boat will have done 24 knots. This means that the distance between the two is

`d^2 = 24^2 + 30^2`

`d = sqrt(1476)`

We know now that `h = 2` and `sqrt(1476)`.

`738(2) = 2sqrt(1476)((dd)/dt)`

`738/sqrt(1476) = (dd)/dt`

`sqrt(1476)/2 = (dd)/dt`

We cannot forget units, which will be knots per hour.

Hopefully this helps!