SOLUTION: A boat is pulled toward dock by means of a rope wound on a drum that is located 6ft above the bow of the boat. if the rope is being pulled in at the rate of 6ft/sec, how fast is th

We can describe the triangle by the Pythagorean theorem:

               
                

We take the derivative implicitly:

                
                   
                    
 
We are told that  (the rope length) is shrinking at the 
constant rate of 6 ft/sec.  Therefore 
(it is negative because  is shrinking).  So we
substitute  for 

                    
                    
                      

Now we want to freeze the motion at the instant when the boat
is 24 feet from the dock.  That is, when x = 24.

Since               

we substitute 

                
                
                   
                  
                   
                  

So we substitute  for  and
 for  in

               
               
               
               
               
               , approximately.

So x is shrinking at the rate of 6.2 ft/sec. Therefore
that is how fast the boat is approaching the dock at
that instant.              

Edwin