Thread: Related Rates Problems

1. A spotlight on the ground shines on the wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.6 m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building?

2. A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m/s, how fast is the boat approaching the dock when it is 8 m from the dock.


So far related rates problems have been pretty easy since we have been using geometric figures. I can't come up with an equation for either of these so I can solve for the change of rate. Can someone help me come up with the equations and show me how to do it in case I get something similar to these in the future?

Originally Posted by FalconPUNCH!

1. A spotlight on the ground shines on the wall 12 m away. If a man 2 m tall walks from the spotlight toward the building at a speed of 1.6 m/s, how fast is the length of his shadow on the building decreasing when he is 4 m from the building?

So far related rates problems have been pretty easy since we have been using geometric figures. I can't come up with an equation for either of these so I can solve for the change of rate. Can someone help me come up with the equations and show me how to do it in case I get something similar to these in the future?

1) using similar triangles, you have

which implies that

and

use the given and evaluate at x=12-4=8 (note that if x=8, then y=3)

for 2) what do you min by "... m higher than..."?

Hello, FalconPUNCH!

You left out a measurement in #2.
. . I'll pick a convenient value . . .

2. A boat is pulled into a dock by a rope attached to the bow of the boat and
passing through a pulley on the dock that is 6 m higher than the bow of the boat.
If the rope is pulled in at a rate of 1 m/s,
how fast is the boat approaching the dock when it is 8 m from the dock?

Code:

                              * P
                          *   |
                 R    *       |
                  *           | 6
              *               |
          *                   |
    B *   *   *   *   *   *   *
                  x

The boat is at , the pulley is at

The length of the rope is: . . and .

From Pythagorus, we have: .

Differerentiate with respect to time: .
. . and we have: .

Can you finish it now?

Oh sorry I left out 1 m. Thanks for your help. I understand now.