tamika wants to use a sheet of fiberboard 19 inches long to create a skateboard ramp with a $20^{\circ}$ angle of elevation from the ground. how high will the ramp rise from the ground at its highest end? round your answer to the nearest tenth of an inch if necessary.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Right Triangle Trigonometry", "Angle of Elevation" ], "new_concepts": [], "current_concepts": [ "Right Triangle Trigonometry", "Angle of Elevation", "Trigonometric Applications" ] } ## Model the physical scenario Using the **Angle of Elevation** and **Right Triangle Trigonometry** knowledge points $$ \begin{aligned} &\theta = 20^\circ \quad \text{(angle of elevation)}\ &\text{Hypotenuse } (c) = 19\text{ inches} \quad \text{(length of the ramp)}\ &\text{Opposite side } (h) = \text{height of the ramp} \end{aligned} $$ ## Set up the trigonometric equation Using the **Right Triangle Trigonometry** knowledge point $$ \begin{aligned} &\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}\ &\sin(20^\circ) = \frac{h}{19} \end{aligned} $$ ## Solve for the height Using the **Right Triangle Trigonometry** knowledge point $$ \begin{aligned} &h = 19 \cdot \sin(20^\circ)\ &\sin(20^\circ) \approx 0.34202\ &h \approx 19 \cdot 0.34202 \approx 6.49838\text{ inches} \end{aligned} $$ ## Round to the nearest tenth Using the **Trigonometric Applications** knowledge point $$ h \approx 6.5\text{ inches} $$ $6.5\text{ inches}$ { "elements": [ { "type": "point", "params": [[0, 0]], "properties": { "name": "A", "color": "#8C55F2", "withLabel": false } }, { "type": "point", "params": [[5.35, 0]], "properties": { "name": "B", "color": "#8C55F2", "withLabel": false } }, { "type": "point", "params": [[5.35, 1.95]], "properties": { "name": "C", "color": "#8C55F2", "withLabel": false } }, { "type": "segment", "params": [[0, 0], [5.35, 0]], "properties": { "strokeColor": "#583C87", "strokeWidth": 2, "name": "Ground" } }, { "type": "segment", "params": [[5.35, 0], [5.35, 1.95]], "properties": { "strokeColor": "#F2557F", "strokeWidth": 2, "name": "h" } }, { "type": "segment", "params": [[0, 0], [5.35, 1.95]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 3, "name": "Ramp (19 in)" } }, { "type": "angle", "params": [[5.35, 0], [0, 0], [5.35, 1.95]], "properties": { "radius": 1, "strokeColor": "#F28A55", "withLabel": true, "name": "20°" } }, { "type": "text", "params": [2.3, 1.3, "19 in"], "properties": { "color": "#5583F2" } }, { "type": "text", "params": [5.6, 1.0, "h"], "properties": { "color": "#F2557F" } } ] } { "subject": "Mathematics", "question_type": "Word Problem", "knowledge_point": [ "Mathematics", "Geometry", "Trigonometric Applications" ] }

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Question

Explanation:

Model the physical scenario

Set up the trigonometric equation

Solve for the height

Round to the nearest tenth

Answer: