from a 40-ft observation tower on the coast, coast guard officer nevinski sights a boat in difficulty. the angles of depression to the boat is $4^{\circ}$. how far is the boat from the shoreline? round to the nearest tenth of a foot.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Trigonometric Ratios", "Angle of Depression" ], "new_concepts": [], "current_concepts": [ "Trigonometric Ratios", "Angle of Depression" ] } ## Set up the trigonometric equation $$ \begin{aligned} &\theta = 4^\circ\ &\text{Opposite} = 40\text{ ft}\ &\tan(4^\circ) = \frac{40}{d} \end{aligned} $$ ## Solve for the distance $$ \begin{aligned} &d = \frac{40}{\tan(4^\circ)}\ &d \approx \frac{40}{0.0699268}\ &d \approx 572.0058\text{ ft} \end{aligned} $$ ## Round to the nearest tenth $$ d \approx 572.0\text{ feet} $$ From a 40-ft observation tower on the coast, Coast Guard Officer Nevinski sights a boat in difficulty. The angles of depression to the boat is $4^\circ$. How far is the boat from the shoreline? Round to the nearest tenth of a foot. 572.0 feet { "elements": [ { "type": "point", "params": [[0, 0]], "properties": { "name": "Shoreline", "size": 3, "color": "#583C87", "withLabel": true } }, { "type": "point", "params": [[0, 4]], "properties": { "name": "Tower Top", "size": 3, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[8, 0]], "properties": { "name": "Boat", "size": 3, "color": "#F2557F", "withLabel": true } }, { "type": "segment", "params": [[0, 0], [0, 4]], "properties": { "strokeColor": "#583C87", "strokeWidth": 3, "name": "40 ft" } }, { "type": "segment", "params": [[0, 0], [8, 0]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2, "name": "d" } }, { "type": "segment", "params": [[0, 4], [8, 0]], "properties": { "strokeColor": "#F28A55", "strokeWidth": 2 } }, { "type": "line", "params": [[0, 4], [4, 4]], "properties": { "strokeColor": "gray", "strokeWidth": 1, "dash": 2 } } ], "timestamps": [0.1, 0.5, 1.0] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Trigonometric Ratios" ] }

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Explanation:

Set up the trigonometric equation

Solve for the distance

Round to the nearest tenth

Answer: