Calculate Rate for

Question

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 1 m higher than the bow of the boat.
If the rope is pulled in at a rate of 1.1 m/s, how fast is the boat approaching the dock when it is 10 m from the dock?

Rate= 

Answer 1

Let x = horizontal distance of the bow of the boat from the dock at time t

 

      y = length of rope at time t

 

We have a right triangle.  At time t, the legs of the triangle have lengths 1 and x and the hypotenuse has length y.

 

By the Pythagorean Theorem, x² + 1² = y²

 

We are given:  dy/dt = -1.1

 

We are to find dx/dt when x = 10 m

 

Differentiate x² + 1 = y² with respect to t:

 

         2x(dx/dt) + 0 = 2y(dy/dt)

 

         So, dx/dt = y(dy/dt)/x        When x = 10, y² = 101

 

                                                                                   y = √101

 

                          = (-1.1)(√101)/10

 

                          = -11(√101)/100

 

When x = 10 m, the boat is approaching the dock at the rate of             11√101/100 m/sec.