Question

from the observation deck of a skyscraper, sophia measures a \\(45^\circ\\) angle of depression to a ship in the harbor below. if the observation deck is 870 feet high, what is the horizontal distance from the base of the skyscraper out to the ship? round your answer to the nearest hundredth of a foot if necessary.

Explanation:

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</pre_analysis>

<reasoning>

Identify the geometric model

\[

$$\begin{aligned} &\theta = 45^\circ \quad \text{(Angle of depression)} \\ &\text{Opposite side} = 870 \text{ ft} \quad \text{(Height of the skyscraper)} \\ &\text{Adjacent side} = x \quad \text{(Horizontal distance to the ship)} \end{aligned}$$

\]

Set up the trigonometric equation

\[
\tan(45^\circ) = \frac{870}{x}
\]

Solve for the horizontal distance

\[

$$\begin{aligned} &1 = \frac{870}{x} \\ &x = 870 \text{ ft} \end{aligned}$$

\]
</reasoning>

<answer>
\(870\)
</answer>

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"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Trigonometric Applications"
]
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</post_analysis>

Answer:

<pre_analysis>
{
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"quick"
],
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],
"new_concepts": [],
"current_concepts": [
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"Trigonometric Applications",
"Angle of Depression"
]
}
</pre_analysis>

<reasoning>

Identify the geometric model

\[

$$\begin{aligned} &\theta = 45^\circ \quad \text{(Angle of depression)} \\ &\text{Opposite side} = 870 \text{ ft} \quad \text{(Height of the skyscraper)} \\ &\text{Adjacent side} = x \quad \text{(Horizontal distance to the ship)} \end{aligned}$$

\]

Set up the trigonometric equation

\[
\tan(45^\circ) = \frac{870}{x}
\]

Solve for the horizontal distance

\[

$$\begin{aligned} &1 = \frac{870}{x} \\ &x = 870 \text{ ft} \end{aligned}$$

\]
</reasoning>

<answer>
\(870\)
</answer>

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"knowledge_point": [
"Mathematics",
"Geometry",
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}
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