a boat is being pulled toward a dock by a rope attached to a winch mounted to the dock. the distance from…

a boat is being pulled toward a dock by a rope attached to a winch mounted to the dock. the distance from the winch to the water is 3 meters. which of the following best describes the relationship between the rate at which the rope is being pulled in, and the rate at which the boat is moving toward the dock? choose one a. the speed at which the rope is pulled in is greater than the speed at which the boat is moving b. the two speeds are the same c. the speed at which the rope is pulled in is less than the speed at which the boat is moving d. there is no relationship between the two speeds

a boat is being pulled toward a dock by a rope attached to a winch mounted to the dock. the distance from the winch to the water is 3 meters. which of the following best describes the relationship between the rate at which the rope is being pulled in, and the rate at which the boat is moving toward the dock? choose one a. the speed at which the rope is pulled in is greater than the speed at which the boat is moving b. the two speeds are the same c. the speed at which the rope is pulled in is less than the speed at which the boat is moving d. there is no relationship between the two speeds

Answer

Explanation:

Step1: Set up variables

Let (x) be the horizontal distance from the boat to the dock, and (l) be the length of the rope. By the Pythagorean theorem, (l^{2}=x^{2}+3^{2}).

Step2: Differentiate with respect to time (t)

Differentiating both sides of (l^{2}=x^{2}+9) with respect to (t) gives (2l\frac{dl}{dt}=2x\frac{dx}{dt}), so (\frac{dl}{dt}=\frac{x}{l}\frac{dx}{dt}). Since (l > x) (because (l=\sqrt{x^{2}+9})), then (\frac{x}{l}<1).

Answer:

C. The speed at which the rope is pulled in is less than the speed at which the boat is moving