a boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the…

Explanation:

Step1: Set up the Pythagorean theorem

Let (x) be the distance of the boat from the dock, and (y) be the length of the rope. By the Pythagorean theorem, (y^{2}=x^{2}+1^{2}=x^{2} + 1).

Step2: Differentiate both sides with respect to time (t)

Differentiating (y^{2}=x^{2}+1) with respect to (t) gives (2y\frac{dy}{dt}=2x\frac{dx}{dt}). Then (\frac{dx}{dt}=\frac{y}{x}\cdot\frac{dy}{dt}).

Step3: Find (y) when (x = 7)

When (x = 7), (y=\sqrt{x^{2}+1}=\sqrt{7^{2}+1}=\sqrt{49 + 1}=\sqrt{50}=5\sqrt{2}).

Step4: Substitute values into (\frac{dx}{dt}) formula

We know that (\frac{dy}{dt}=- 1) (negative because (y) is decreasing). Substituting (x = 7), (y = 5\sqrt{2}), and (\frac{dy}{dt}=-1) into (\frac{dx}{dt}=\frac{y}{x}\cdot\frac{dy}{dt}), we get (\frac{dx}{dt}=\frac{5\sqrt{2}}{7}\times(-1)\approx - 1.01). The negative sign indicates the boat is approaching the dock.

Answer:

(1.01)