Question

Hauling in a dinghy A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft above the bow. The rope is hauled in at the rate of 2 ft/sec.

a. How fast is the boat approaching the dock when 10 ft of rope are out?

Diagram showing a dinghy being pulled toward a dock by a rope through a ring 6 ft above the water.

Diagram showing a dinghy being pulled toward a dock by a rope through a ring 6 ft above the water.

Accepted Answer

Identify the right triangle formed by the rope, the vertical distance from the dock to the bow (6 ft), and the horizontal distance from the dock to the boat. Let x be the horizontal distance from the dock to the boat, and let L be the length of the rope. Use the Pythagorean theorem to relate x, L, and the vertical distance: $$ L^2 = x^2 + 6^2 . $$Differentiate both sides of the equation with respect to time t to find the relationship between the rates of change: $$ 2L \frac{dL}{dt} = 2x \frac{dx}{dt} . $$Substitute the given values into the differentiated equation. You know $$ \frac{dL}{dt} = -2  ft/sec ($$since the rope is being pulled in), L = 10 ft, and solve for $$ \frac{dx}{dt} $$, the rate at which the boat is approaching the dock. Calculate x when L = 10 ft using the Pythagorean theorem: $$ x = \sqrt{L^2 - 6^2} . $$Substitute this value of x into the differentiated equation to find $$ \frac{dx}{dt} .$$

Hauling in a dinghy A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft above the bow. The rope is hauled in at the rate of 2 ft/sec.

a. How fast is the boat approaching the dock when 10 ft of rope are out?

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Key Concepts

Related Rates

Pythagorean Theorem

Differentiation

Hauling in a dinghy A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft above the bow. The rope is hauled in at the rate of 2 ft/sec.a. How fast is the boat approaching the dock when 10 ft of rope are out?