4. a cliff casts a shadow that is 12 metres long when the sun is shining at an angle of $78^\circ$ to the horizontal. later in the day, the shadow is 12 metres longer. at what angle to the ground is the sun now shining?
hint: using the small triangle, find the height of the cliff, then from the large triangle find the size of the angle.

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Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Right Triangle Trigonometry", "Inverse Trigonometric Functions" ], "new_concepts": [], "current_concepts": [ "Right Triangle Trigonometry", "Inverse Trigonometric Functions" ] } ## Find the height of the cliff Using the *Right Triangle Trigonometry* knowledge point $$ \begin{aligned} &\tan(78^\circ) = \frac{h}{12} \ &h = 12 \cdot \tan(78^\circ) \ &h \approx 12 \cdot 4.7046 = 56.455\text{ m} \end{aligned} $$ ## Determine the total shadow length Using the *Right Triangle Trigonometry* knowledge point $$ \begin{aligned} &L = 12 + 12 = 24\text{ m} \end{aligned} $$ ## Calculate the new angle of elevation Using the *Inverse Trigonometric Functions* knowledge point $$ \begin{aligned} &\tan(\theta) = \frac{h}{24} \ &\tan(\theta) = \frac{12 \cdot \tan(78^\circ)}{24} = \frac{\tan(78^\circ)}{2} \ &\tan(\theta) \approx \frac{4.7046}{2} = 2.3523 \ &\theta = \arctan(2.3523) \approx 67^\circ \end{aligned} $$ $67^\circ$ { "elements": [ { "type": "point", "params": [[0, 0]], "properties": { "name": "Base", "size": 3, "color": "#583C87", "withLabel": false } }, { "type": "point", "params": [[0, 5.65]], "properties": { "name": "Cliff Top", "size": 3, "color": "#583C87", "withLabel": false } }, { "type": "point", "params": [[1.2, 0]], "properties": { "name": "Shadow 1", "size": 3, "color": "#583C87", "withLabel": false } }, { "type": "point", "params": [[2.4, 0]], "properties": { "name": "Shadow 2", "size": 3, "color": "#583C87", "withLabel": false } }, { "type": "segment", "params": [[0, 0], [0, 5.65]], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "name": "h" } }, { "type": "segment", "params": [[0, 0], [2.4, 0]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 3 } }, { "type": "segment", "params": [[0, 5.65], [1.2, 0]], "properties": { "strokeColor": "#55DDF2", "strokeWidth": 2 } }, { "type": "segment", "params": [[0, 5.65], [2.4, 0]], "properties": { "strokeColor": "#F2557F", "strokeWidth": 2 } }, { "type": "text", "params": [-0.5, 2.8, "h \approx 56.5\text{ m}"], "properties": { "color": "#8C55F2" } }, { "type": "text", "params": [0.5, -0.4, "12\text{ m}"], "properties": { "color": "#5583F2" } }, { "type": "text", "params": [1.7, -0.4, "12\text{ m}"], "properties": { "color": "#5583F2" } }, { "type": "text", "params": [0.8, 0.3, "78^\circ"], "properties": { "color": "#55DDF2" } }, { "type": "text", "params": [2.0, 0.3, "\theta \approx 67^\circ"], "properties": { "color": "#F2557F" } } ] } { "subject": "Mathematics", "question_type": "Word Problem", "knowledge_point": [ "Mathematics", "Geometry", "Right Triangle Trigonometry" ] }

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Explanation:

Find the height of the cliff

Determine the total shadow length

Calculate the new angle of elevation

Answer: