Question

from a hot-air balloon, khalil measures a \\(36^\circ\\) angle of depression to a landmark thats 1304 feet away, measuring horizontally. whats the balloons vertical distance above the ground? round your answer to the nearest hundredth of a foot if necessary.

Explanation:

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<reasoning>

Model the physical scenario

Using the Trigonometric Applications knowledge point, we can represent this situation with a right triangle. Let:

• \(h\) be the vertical height of the hot-air balloon above the ground.
• \(d = 1304\text{ ft}\) be the horizontal distance from the balloon to the landmark.
• \(\theta = 36^\circ\) be the angle of depression.

By alternate interior angles, the angle of elevation from the landmark to the balloon is also \(36^\circ\).

Set up the trigonometric equation

Using the Right Triangle Trigonometry knowledge point, we relate the opposite side \(h\) and the adjacent side \(d\) to the angle \(\theta\):
\[
\tan(36^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{1304}
\]

Solve for the vertical distance

Isolate \(h\) by multiplying both sides by \(1304\):
\[
h = 1304 \cdot \tan(36^\circ)
\]

Calculate the numerical value

Using a calculator to find \(\tan(36^\circ) \approx 0.7265425\):
\[
h \approx 1304 \cdot 0.7265425 \approx 947.4114\text{ ft}
\]

Round to the nearest hundredth

Rounding \(947.4114\) to two decimal places gives:
\[
h \approx 947.41\text{ ft}
\]
</reasoning>

<answer>
\(947.41\text{ feet}\)
</answer>

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{
"type": "point",
"params": [[0, 0]],
"properties": {
"name": "Landmark",
"size": 4,
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Answer:

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<reasoning>

Model the physical scenario

Using the Trigonometric Applications knowledge point, we can represent this situation with a right triangle. Let:

• \(h\) be the vertical height of the hot-air balloon above the ground.
• \(d = 1304\text{ ft}\) be the horizontal distance from the balloon to the landmark.
• \(\theta = 36^\circ\) be the angle of depression.

By alternate interior angles, the angle of elevation from the landmark to the balloon is also \(36^\circ\).

Set up the trigonometric equation

Using the Right Triangle Trigonometry knowledge point, we relate the opposite side \(h\) and the adjacent side \(d\) to the angle \(\theta\):
\[
\tan(36^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{1304}
\]

Solve for the vertical distance

Isolate \(h\) by multiplying both sides by \(1304\):
\[
h = 1304 \cdot \tan(36^\circ)
\]

Calculate the numerical value

Using a calculator to find \(\tan(36^\circ) \approx 0.7265425\):
\[
h \approx 1304 \cdot 0.7265425 \approx 947.4114\text{ ft}
\]

Round to the nearest hundredth

Rounding \(947.4114\) to two decimal places gives:
\[
h \approx 947.41\text{ ft}
\]
</reasoning>

<answer>
\(947.41\text{ feet}\)
</answer>

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