Final answer:
To determine how fast the boat is approaching the dock, use the Pythagorean theorem to relate the sides of the triangle formed by the boat, dock, and pulley. Differentiate this relationship with respect to time to find the rate at which the boat's distance from the dock is changing. When the boat is 8 meters away, it is found to approach the dock at a rate of 3.75 meters/sec.
Step-by-step explanation:
The question involves using rates of change to find how fast the boat is being pulled towards a dock by a rope that is passing through a pulley on the dock, which is positioned 6 meters higher than the boat.
We can treat this scenario as a right triangle where the vertical side represents the height of the pulley above the boat (6 meters), the horizontal side is the distance of the boat from the dock (8 meters) and the hypotenuse is the length of the rope from the boat to the pulley. As the rope's length shortens at a rate of 3 meters/sec, we are looking for the rate at which the boat approaches the dock (dx/dt), where x is the distance from the boat to the dock.
Step 1: Express the relationship using Pythagoras' theorem
The relationship between the distance from the boat to the dock (x), the fixed height of the pulley (6 meters), and the length of the rope (l) is given by:
l^2 = x^2 + 6^2
Step 2: Differentiate with respect to time
Using calculus, differentiate both sides of the equation with respect to time (t):
2l*(dl/dt) = 2x*(dx/dt)
Since the rope is shortening, dl/dt is -3 meters/sec (the negative sign indicates a decrease in length).
Step 3: Plug in the values and solve for dx/dt
When x = 8 meters:
2l*(-3) = 2*8*(dx/dt)
First, we need to find the length of the rope (l) when the boat is 8 meters from the dock using the Pythagorean theorem:
l = √(8^2 + 6^2) = √(64 + 36) = √100 = 10 meters
Now we substitute the values:
-60 = 16*(dx/dt)
(dx/dt) = -60 / 16 = -3.75 meters/sec
The negative sign indicates that the boat is moving towards the dock.