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

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 m/s, how fast is the boat approaching the dock when it is 6 m from the dock?

Answer 1

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

     y = length of rope at time t

 

Draw a right triangle with horizontal leg x, vertical leg 1, and hypotenuse y.

 

We know that dy/dt = -1 and want to find dx/dt when x = 6.

 

By the Pythagorean Theorem, x² + 1² = y².

 

    2x(dx/dt) = 2y(dy/dt)

 

         dx/dt = y(dy/dt) / x     When x = 6, y² = 36 + 1 = 37

 

                                                             So, y = √37

 

          dx/dt = -√37 / 6

 

When the boat is 6 m from the dock, it is approaching the dock at the rate of √37/6 m/sec.

 

Answer 2

boat-pulley form a right triangle with legs 1 and x 

where x is the distance from the boat to dock.

 

The hypotenuse h(x) = sqrt(x^2+1)

 

dH/dx = 2x/ ( x^2+1)^(1/2)

 

 

1 = (2*6)/(6^2+1)^(1/2) dx/dt

 

1 = [(12)/sqrt(37)] (dx/dt)

 

sqrt(37)/12 = dx/dt