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.

b. At what rate is the angle θ changing at this instant (see the figure)?
A dinghy being pulled toward a dock by a rope, with a ring on the dock 6 ft above the bow, illustrating related rates.

A dinghy being pulled toward a dock by a rope, with a ring on the dock 6 ft above the bow, illustrating related rates.

Accepted Answer

First, identify the relationship between the variables involved. We have a right triangle where the vertical side is 6 ft (the height of the ring above the bow), the hypotenuse is the length of the rope, and the angle θ is formed between the rope and the horizontal line from the bow to the dock. Use trigonometric relationships to express the angle θ in terms of the length of the rope. Specifically, use the sine function: sin(θ) = opposite/hypotenuse = 6/L, where L is the length of the rope. Differentiate the equation sin(θ) = 6/L with respect to time t to find the rate of change of the angle θ. This involves implicit differentiation. Remember that dL/dt is given as -2 ft/sec (since the rope is being pulled in, the length is decreasing). Apply the chain rule to differentiate: d(sin(θ))/dt = cos(θ) * dθ/dt and d(6/L)/dt = $$-6/L^2 * dL/dt. $$Set these two derivatives equal to each other: cos(θ) * dθ/dt = $$-6/L^2 * (-2). $$Solve for dθ/dt, the rate at which the angle θ is changing. Substitute the known values into the equation, including the current length of the rope if needed, to find the expression for dθ/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.

b. At what rate is the angle θ changing at this instant (see the figure)?

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Key Concepts

Related Rates

Trigonometric Relationships

Implicit 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.b. At what rate is the angle θ changing at this instant (see the figure)?