Final answer:
The boat is approaching the dock at a rate of 6 ft/sec when 19 ft of rope are out.
Step-by-step explanation:
In this problem, we can use the concept of related rates to find the boat's speed when a certain length of rope is out. We have a right triangle formed by the boat, the rope, and the vertical distance between the bow and the ring on the dock.
The length of the rope forms one leg of the triangle, and the vertical distance forms the other leg. The boat's speed is represented by the hypotenuse of the triangle.
We are given that the rope is hauled in at a rate of 3 ft/sec, so its length is changing over time. We need to find the boat's speed, which is the rate at which the hypotenuse is changing when 19 ft of rope are out.
To solve this problem, we can use the Pythagorean theorem to relate the lengths of the sides of the triangle. Let's define the length of the rope as 'r', and the vertical distance as 'h'.
According to the Pythagorean theorem, r^2 = h^2 + 9^2. Differentiating both sides of this equation with respect to time gives 2r * dr/dt = 2h * dh/dt.
Since we know the rate at which the rope is being hauled in (dr/dt = 3 ft/sec) and we want to find the rate at which the boat is approaching the dock (dh/dt), we can substitute these values into the equation. Let's solve for dh/dt:
2r * 3 ft/sec = 2h * dh/dt. We can simplify this equation by canceling out the common factors of 2, giving us:
3r = h * dh/dt. Finally, we can plug in the given values of r = 19 ft and solve for dh/dt:
3 * 19 ft = 9 ft * dh/dt.
57 ft = 9 ft * dh/dt.
dh/dt = 57 ft / 9 ft.
dh/dt = 6 ft/sec.
Therefore, the boat is approaching the dock at a rate of 6 ft/sec when 19 ft of rope are out.