A boat is pulled

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 7 m from the dock? (Round your answer to two decimal places.)

Answer 1

The boat is approaching the dock at approximately 1.01 m/s.

We can use the Pythagorean theorem since the rope, the horizontal distance to the dock, and the 1-meter height difference form a right triangle.

Let x be the boat's distance from the dock and y be the length of the rope.

x² + 1 = y²

Differentiate both sides with respect to time:

2x(dx/dt) = 2y(dy/dt)

Since the rope is being pulled in at 1 m/s, dy/dt = -1.

When x = 7, the rope length is:

y = sqrt(49 + 1) = sqrt(50)

Substituting:

7(dx/dt) = sqrt(50)(-1)

dx/dt = -1.01 m/s

The negative sign means the distance is decreasing, so the boat is approaching the dock at 1.01 m/s.