from a hot-air balloon, violet measures a $27^{\circ}$ angle of depression to a landmark thats 790 feet away, measuring horizontally. whats the balloons vertical distance above the ground? round your answer to the nearest tenth of a foot if necessary.

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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": true, "concepts_used": [ "Trigonometric Applications", "Right Triangle Trigonometry" ], "new_concepts": [], "current_concepts": [ "Trigonometric Applications", "Right Triangle Trigonometry", "Angle of Depression" ] } ## Identify the geometric relationship Using the **Trigonometric Applications** and **Right Triangle Trigonometry** knowledge points $$ \begin{aligned} &\theta = 27^\circ \quad \text{(angle of depression / elevation)}\ &d = 790\text{ ft} \quad \text{(horizontal distance)}\ &h = \text{vertical height above the ground} \end{aligned} $$ ## Set up the trigonometric equation Using the **Right Triangle Trigonometry** knowledge point $$ \begin{aligned} &\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\ &\tan(27^\circ) = \frac{h}{790} \end{aligned} $$ ## Solve for the vertical distance Using the **Trigonometric Applications** knowledge point $$ \begin{aligned} &h = 790 \cdot \tan(27^\circ)\ &\tan(27^\circ) \approx 0.509525\ &h \approx 790 \cdot 0.509525 \approx 402.525\text{ ft}\ &h \approx 402.5\text{ ft} \end{aligned} $$ From a hot-air balloon, Violet measures a $27^\circ$ angle of depression to a landmark that's 790 feet away, measuring horizontally. What's the balloon's vertical distance above the ground? Round your answer to the nearest tenth of a foot if necessary. 402.5 feet { "elements": [ { "type": "point", "params": [[0, 0]], "properties": { "name": "Landmark", "size": 4, "color": "#583C87", "withLabel": true } }, { "type": "point", "params": [[7.9, 0]], "properties": { "name": "Ground Point", "size": 3, "color": "#5583F2", "withLabel": false } }, { "type": "point", "params": [[7.9, 4.03]], "properties": { "name": "Balloon", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "segment", "params": [[0, 0], [7.9, 0]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2, "name": "790 ft", "withLabel": true } }, { "type": "segment", "params": [[7.9, 0], [7.9, 4.03]], "properties": { "strokeColor": "#F2557F", "strokeWidth": 2, "name": "h", "withLabel": true } }, { "type": "segment", "params": [[0, 0], [7.9, 4.03]], "properties": { "strokeColor": "#55DDF2", "strokeWidth": 2 } }, { "type": "line", "params": [[7.9, 4.03], [4.0, 4.03]], "properties": { "strokeColor": "gray", "strokeWidth": 1, "dash": 2 } }, { "type": "angle", "params": [[4.0, 4.03], [7.9, 4.03], [0, 0]], "properties": { "name": "27°", "withLabel": true } } ] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Trigonometric Applications" ] }

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Explanation:

Identify the geometric relationship

Set up the trigonometric equation

Solve for the vertical distance

Answer: