Questions: A ship's sonar finds that the angle of depression to a wreck on the bottom of the ocean is 12.5 degrees. If a point on the ocean floor is 60 meters directly below the ship, how many meters is it from that point on the ocean floor to the wreck? Round your answer to the nearest tenth. (A) 277.2 m (B) 270.6 m (C) 61.5 m (D) 13.3 m Which triangle should be solved by beginning with the Law of Cosines? (A) m angle A=115, a=19, b=13

Solution Steps

We are given a 30-60-90 right triangle. The side opposite the 60° angle has length 5. We are asked to find the length of the hypotenuse, denoted by x.

In a 30-60-90 triangle, the side lengths are in the ratio 1:√3:2. The side opposite the 30° angle is the shortest side, the side opposite the 60° angle is the middle side, and the hypotenuse is the longest side. Since the side opposite the 60° angle is 5, we can set up the proportion: 5 / √3 = x / 2

Cross-multiply to get: 5 * 2 = x * √3 10 = x√3 x = 10/√3

Multiply the numerator and denominator by √3 to get: x = (10√3) / 3

Since none of the provided options are equivalent to $\frac{10\sqrt{3}}{3}$, there seems to be a typo in the problem, but none of the options are correct. If the side opposite the 60 degree angle was instead adjacent to the 60 degree angle, the hypotenuse would be 10.

We are given an angle of depression of 12.5° and a vertical distance of 60 meters. We want to find the distance from the point below the ship to the wreck. This forms a right triangle where the vertical distance is the adjacent side to the angle of depression.

Let d be the horizontal distance to the wreck. We have: tan(12.5°) = d / 60. So, d = 60 * tan(12.5°) ≈ 13.3 m.

The Law of Cosines should be used when you have Side-Side-Side (SSS) or Side-Angle-Side (SAS). Option (A) gives an angle and two sides, but the angle is not between the given sides. This is the Side-Side-Angle (SSA) case, also known as the ambiguous case, where the Law of Sines should be applied first.

Final Answer