Thread: least distance

1.) A ship lies 6 mi. from shore, and opposite a point 10 mi. farther along the shore another ship lies 18 mi. offshore. A boat from the first ship is to land a passenger and then proceed to the other ship. What is the least distance the boat can travel?

Help pls...can anyone show me the diagram of this problem.


-tnx

Hello, ihmth!

There is a back-door approach to this problem,
. . involving optics and striaght-line paths.
But I'll use the expected Calculus set-up.

1) A ship lies 6 mi. from shore.
Opposite a point 10 mi. farther along the shore another ship lies 18 mi. offshore.
A boat from the first ship is to land a passenger and then proceed to the other ship.
What is the least distance the boat can travel?

Code:

                                      * C
                                    * |
                                  *   |
                                *     |
    A *                       *       |
      | *                   *         | 16
      |   *               *           |
    6 |     *           *             |
      |       *       *               |
      |         *   *                 |
    B * - - - - - * - - - - - - - - - * D
            x     P       10 - x

The first ship is

The second ship is at
And: .

Let be the landing point.
Then: .

In right triangle

In right triangle

The distance travelled is: .

. . and that is the function we must minimize.