Question

A boat is parked at the end of a 20 ft dock. The boat is tied to a pole at the opposite end of the dock with a 35 foot rope. What is the angle formed by the rope and the dock? Round your answer to the nearest degree

Answer 1

Answer:

Approximately $55^{\circ}$.

Step-by-step explanation:

Refer to the diagram attached. In the diagram, the horizontal line segment represent the dock, and the slanted line segment represents the rope. Given that the boat is parked at the other side of the dock, the three line segments form a right triangle:

• The dock,
• The rope, as well as
• The line segment between the boat and the other side of the dock.

The question is asking for the angle $\theta$ between the dock and the rope. The length of both the dock (the side adjacent to the angle) and the rope (the hypotenuse) are given. The ratio between the two sides would be equal to the cosine of this angle:

$\begin{aligned}\cos({\rm \theta}) = \frac{(\text{adjacent})}{(\text{hypotenuse})} = \frac{20\; {\rm ft}}{35\; {\rm ft}}\end{aligned}$.

Hence, the angle between the dock and the rope would be:

$\begin{aligned}\theta &= \arccos\left(\frac{20\; {\rm ft}}{35\; {\rm ft}}\right) \approx 55^{\circ}\end{aligned}$.