Question

Question #aac39

Answer 1

`5sqrt(113) ~~ 53.15mph`

Explanation:

The motion of the two boats describes a right triangle with the right angle in the lower left at the origin, and one boat moving up the y-axis with the other boat moving to the right on the x-axis.

To find a relationship regarding the distance between the two boats, we can recognize that this distance is the hypotenuse of the right triangle. Thus, a common expression regarding right triangles is the Pythagorean Formula:

`a^2 + b^2 = c^2`

If we consider the eastward boat as `a` and the northward boat as `b`, we can consider the hypotenuse as being `c`.

You can differentiate the previous formula with respect to time `t` using the Chain Rule:

`2a*(da)/(dt) + 2b*(db)/(dt) = 2c*(dc)/(dt)`

`a*(da)/(dt) + b*(db)/(dt) = c*(dc)/(dt) color(white)("aaaaa")[1]`

We know the eastward boat is traveling at 40mph, so `(da)/(dt) = 40`. We also know that 2 hours have passed, and so we know `a = 80`

As for the northward boat, it is traveling at 35mph, so `(db)/(dt) = 35`, and after 2 hours have passed, `b = 70`.

Using the Pythagorean Formula, we can find the hypotenuse length `c` at this point in time:

`c^2 = a^2 + b^2 = 80^2 + 70^2 = 6400 + 4900 = 11300`

`c = sqrt(11300) = 10sqrt(113)`

We now have everything we need to substitute into [1] above and solve for the unknown `(dc)/(dt)`:

`a*(da)/(dt) + b*(db)/(dt) = c*(dc)/(dt)`

`(80)(40) + (70)(35) = 10sqrt(113)*(dc)/(dt)`

`3200 + 2450 = 10sqrt(113)*(dc)/(dt)`

`5650 = 10sqrt(113)*(dc)/(dt)`

`5650/(10sqrt(113)) = (dc)/(dt)`

`(dc)/(dt) = 565/sqrt(113) = (565sqrt(113))/113 = 5sqrt(113)`

Aside: With some tedious work, you can prove that in problems like this, the rate of change of the hypotenuse can be found by just applying the Pythagorean Formula to the two boats' speeds without worrying about how far they've traveled.