a pilot is flying over the ocean. he determines that the angles of depression to two ships are $62^\circ$ and $55^\circ$, as shown in the figure below. the plane is $5\text{ miles}$ from the ship located at point $a$. how far apart are the ships? round your answer to the nearest tenth of a mile.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Solving Word Problems with Triangles", "Angle Sum Theorem", "Law of Sines" ], "new_concepts": [], "current_concepts": [ "Solving Word Problems with Triangles", "Angle Sum Theorem", "Law of Sines" ] } ## Find the interior angles of the triangle Using the *Solving Word Problems with Triangles* and *Angle Sum Theorem* knowledge points $$ \begin{aligned} &\angle A = 55^\circ \quad \text{(alternate interior angles)} \ &\angle B = 62^\circ \quad \text{(alternate interior angles)} \ &\angle P = 180^\circ - (55^\circ + 62^\circ) = 63^\circ \end{aligned} $$ ## Apply the Law of Sines to find the distance between the ships Using the *Law of Sines* knowledge point $$ \begin{aligned} &\frac{AB}{\sin(63^\circ)} = \frac{5}{\sin(62^\circ)} \ &AB = \frac{5 \cdot \sin(63^\circ)}{\sin(62^\circ)} \ &AB \approx \frac{5 \cdot 0.8910}{0.8829} \approx 5.0457 \text{ miles} \end{aligned} $$ ## Round the final answer to the nearest tenth Using the *Solving Word Problems with Triangles* knowledge point $$ AB \approx 5.0 \text{ miles} $$ A pilot is flying over the ocean. He determines that the angles of depression to two ships are $62^\circ$ and $55^\circ$, as shown in the figure below. The plane is $5\text{ miles}$ from the ship located at point $A$. How far apart are the ships? Round your answer to the nearest tenth of a mile. 5.0 miles { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Solving Word Problems with Triangles" ] }

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Question

Explanation:

Find the interior angles of the triangle

Apply the Law of Sines to find the distance between the ships

Round the final answer to the nearest tenth

Answer: