as shown in the figure below, alonzo is standing 91 feet from the base of a leaning tree. the tree is growing at an angle of $79^\circ$ with respect to the ground. the angle of elevation from where alonzo is standing to the top of the tree is $41^\circ$. find the length, $x$, of the tree. round your answer 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", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Law of Sines", "Angle Sum Theorem" ], "new_concepts": [ "Solving Word Problems with Triangles" ], "current_concepts": [ "Law of Sines", "Angle Sum Theorem", "Solving Triangles" ] } ## Identify the given parts of the triangle Using the *Angle Sum Theorem* and *Law of Sines* knowledge points * Let $A$ be the angle where Alonzo is standing: $A = 41^\circ$. * Let $B$ be the angle between the ground and the leaning tree: $B = 79^\circ$. * Let $c$ be the distance from Alonzo to the base of the tree: $c = 91\text{ ft}$. * Let $C$ be the angle at the top of the tree. * Let $a = x$ be the length of the tree opposite angle $A$. ## Find the third angle of the triangle Using the *Angle Sum Theorem* knowledge point $$ \begin{aligned} C &= 180^\circ - A - B \ C &= 180^\circ - 41^\circ - 79^\circ \ C &= 60^\circ \end{aligned} $$ ## Apply the Law of Sines to find x Using the *Law of Sines* knowledge point $$ \begin{aligned} \frac{x}{\sin(A)} &= \frac{c}{\sin(C)} \ \frac{x}{\sin(41^\circ)} &= \frac{91}{\sin(60^\circ)} \ x &= \frac{91 \cdot \sin(41^\circ)}{\sin(60^\circ)} \end{aligned} $$ ## Calculate the numerical value of x Using the *Law of Sines* knowledge point $$ \begin{aligned} \sin(41^\circ) &\approx 0.656059 \ \sin(60^\circ) &\approx 0.8660254 \ x &\approx \frac{91 \cdot 0.656059}{0.8660254} \ x &\approx \frac{59.70137}{0.8660254} \approx 68.9372 \approx 68.9\text{ ft} \end{aligned} $$ As shown in the figure below, Alonzo is standing 91 feet from the base of a leaning tree. The tree is growing at an angle of $79^\circ$ with respect to the ground. The angle of elevation from where Alonzo is standing to the top of the tree is $41^\circ$. Find the length, $x$, of the tree. Round your answer to the nearest tenth of a foot. $68.9$ feet { "elements": [ { "type": "point", "params": [[-4.5, -2]], "properties": { "name": "Alonzo (A)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[4.5, -2]], "properties": { "name": "Base (B)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "point", "params": [[3.18, 4.89]], "properties": { "name": "Top (C)", "size": 4, "color": "#8C55F2", "withLabel": true } }, { "type": "segment", "params": [[-4.5, -2], [4.5, -2]], "properties": { "strokeColor": "#583C87", "strokeWidth": 3, "name": "91 ft", "withLabel": true, "labelPosition": [0, -10] } }, { "type": "segment", "params": [[4.5, -2], [3.18, 4.89]], "properties": { "strokeColor": "#F2557F", "strokeWidth": 3, "name": "x", "withLabel": true, "labelPosition": [10, 0] } }, { "type": "segment", "params": [[-4.5, -2], [3.18, 4.89]], "properties": { "strokeColor": "#5583F2", "strokeWidth": 2 } }, { "type": "angle", "params": [[4.5, -2], [-4.5, -2], [3.18, 4.89]], "properties": { "name": "41°", "withLabel": true, "radius": 1.2 } }, { "type": "angle", "params": [[3.18, 4.89], [4.5, -2], [-4.5, -2]], "properties": { "name": "79°", "withLabel": true, "radius": 1.0 } } ], "timestamps": [0.5, 1.0, 1.5] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Law of Sines" ] }

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Question

Explanation:

Identify the given parts of the triangle

Find the third angle of the triangle

Apply the Law of Sines to find x

Calculate the numerical value of x

Answer: