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

Answer 1

To find the speed at which the boat is approaching the dock, we can use the concept of related rates. The distance between the boat and the dock is decreasing, so we need to find the rate at which this distance is changing.

Let's call the distance between the boat and the dock "x" (in meters). The height of the pulley is 1 meter higher than the bow of the boat, so the height of the pulley from the ground is x + 1 meters.

Using the Pythagorean theorem, we can relate the distance x and the height x + 1:

x^2 + 1^2 = (x + 1)^2

Simplifying this equation, we get:

x^2 + 1 = x^2 + 2x + 1

Subtracting x^2 from both sides, we have:

1 = 2x

Dividing both sides by 2, we find:

x = 1/2 meters

Now, let's differentiate both sides of the equation with respect to time:

d(x)/dt = d(1/2)/dt

The rate at which the distance x is changing (d(x)/dt) is the speed at which the boat is approaching the dock. The rate at which the rope is being pulled in (d(1/2)/dt) is given as 1 m/s.

Therefore, the boat is approaching the dock at a speed of 1 m/s when it is 9 meters away.

Hope that helps! Let me know if you have any other questions.