Question

part iii
answer all 3 questions in this part. each correct answer will receive 4 credits. clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. utilize the information provided for each question to determine your answer. note that diagrams are not necessarily drawn to scale. for all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. all answers should be written in pen, except for graphs and drawings, which should be done in pencil. 12

32 as modeled below, maria wants to determine the height of the building across the street from her position, \\(m\\). the angle of elevation from \\(m\\) to the top of the building, \\(t\\), is \\(36^\circ\\). from \\(m\\), the angle of depression to the base of the building, \\(b\\), is \\(18^\circ\\). the buildings are 80 feet apart.

determine and state, to the nearest foot, the height of the building, \\(tb\\), across the street from maria.

Explanation:

<pre_analysis>
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"question_count": 1,
"skills_matched": [
"step_cot",
"explain_then_answer",
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],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Right Triangle Trigonometry",
"Angles of Elevation and Depression"
],
"current_concepts": [
"Right Triangle Trigonometry",
"Angles of Elevation and Depression"
]
}
</pre_analysis>

<reasoning>

Calculate the height of the upper section TH

\[

$$\begin{aligned} &MH = 80\text{ ft} \\ &\tan(36^\circ) = \frac{TH}{MH} = \frac{TH}{80} \\ &TH = 80 \cdot \tan(36^\circ) \approx 58.1234\text{ ft} \end{aligned}$$

\]

Calculate the height of the lower section HB

\[

$$\begin{aligned} &\tan(18^\circ) = \frac{HB}{MH} = \frac{HB}{80} \\ &HB = 80 \cdot \tan(18^\circ) \approx 25.9936\text{ ft} \end{aligned}$$

\]

Calculate the total height TB and round to the nearest foot

\[

$$\begin{aligned} &TB = TH + HB \\ &TB \approx 58.1234 + 25.9936 = 84.1170\text{ ft} \\ &TB \approx 84\text{ ft} \end{aligned}$$

\]
</reasoning>

<answer>
\(84\) feet
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Geometry",
"Right Triangle Trigonometry"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Right Triangle Trigonometry",
"Angles of Elevation and Depression"
],
"current_concepts": [
"Right Triangle Trigonometry",
"Angles of Elevation and Depression"
]
}
</pre_analysis>

<reasoning>

Calculate the height of the upper section TH

\[

$$\begin{aligned} &MH = 80\text{ ft} \\ &\tan(36^\circ) = \frac{TH}{MH} = \frac{TH}{80} \\ &TH = 80 \cdot \tan(36^\circ) \approx 58.1234\text{ ft} \end{aligned}$$

\]

Calculate the height of the lower section HB

\[

$$\begin{aligned} &\tan(18^\circ) = \frac{HB}{MH} = \frac{HB}{80} \\ &HB = 80 \cdot \tan(18^\circ) \approx 25.9936\text{ ft} \end{aligned}$$

\]

Calculate the total height TB and round to the nearest foot

\[

$$\begin{aligned} &TB = TH + HB \\ &TB \approx 58.1234 + 25.9936 = 84.1170\text{ ft} \\ &TB \approx 84\text{ ft} \end{aligned}$$

\]
</reasoning>

<answer>
\(84\) feet
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Geometry",
"Right Triangle Trigonometry"
]
}
</post_analysis>