Thread: Boats?

A boat leaves a dock at 3:00 P.M. and travels due south at a speed of 20 km/h. Another boat has been heading due east at 15 km/h and reaches the same dock at 4:00 P.M. How many minutes after 3:00 P.M. were the two boats closest together?

Originally Posted by sheva2291

A boat leaves a dock at 3:00 P.M. and travels due south at a speed of 20 km/h. Another boat has been heading due east at 15 km/h and reaches the same dock at 4:00 P.M. How many minutes after 3:00 P.M. were the two boats closest together?

Start by setting up an appropriate coordinate system. It will be simplest to set the origin at the dock, the positive x-axis west and the positive y axis south with x and y in km from the dock. Take t in hours with t= 0 at 3:00 P.lM.

The first boat travels due south so its x coordinate is always 0. Its speed is 20 km/h so its distance from the dock, the y coordinate, after t hours is 20t.

The second boat travels due east so its y coordinate is always 0. its speed is -15 (because the positive x-axis points west and the boat is going east) so x= -15t+ b. You can find b by using the fact that the boat "reaches the same dock at 4:00 P.M."


The distance between the two boats is . You can find the minimum distance by differentiating with respect to t and setting the derivative equal to 0 or, for this simple problem, by completing the square.

Hello, sheva2291!

A boat leaves a dock at 3:00 PM and travels due south at 20 km/hr.
Another boat has been heading due east at 15 km/h and reaches the same dock at 4:00 PM.
How many minutes after 3:00 PM were the two boats closest together?

Code:

      : - - -  15 - - - - :
      P  15t  B   15-15t  D
      o-------o-----------o
                *         |
                  *       |
                 x  *     | 20t
                      *   |
                        * |
                          o A

Ship starts at dock and sails south at 20 km/hr,
In hours, it has sailed km to point .

Ship starts at (15 km west of ) and sails east at 15 km/hr.
In hours, it has sailed km to point .
. . Hence: .

Let

In right triangle

. . Hence, we have: .

And that is the function we must minimize.


I believe the answer is: .

ahhh thank you very much, you explained it very clearly. i changed around the numbers and got it right =)