Question

from a hot-air balloon, violet measures a \\(27^{\circ}\\) angle of depression to a landmark thats 790 feet away, measuring horizontally. whats the balloons vertical distance above the ground? round your answer to the nearest tenth of a foot if necessary.

Explanation:

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<reasoning>

Identify the geometric relationship

Using the Trigonometric Applications and Right Triangle Trigonometry knowledge points
\[

$$\begin{aligned} &\theta = 27^\circ \quad \text{(angle of depression / elevation)}\\ &d = 790\text{ ft} \quad \text{(horizontal distance)}\\ &h = \text{vertical height above the ground} \end{aligned}$$

\]

Set up the trigonometric equation

Using the Right Triangle Trigonometry knowledge point
\[

$$\begin{aligned} &\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\\ &\tan(27^\circ) = \frac{h}{790} \end{aligned}$$

\]

Solve for the vertical distance

Using the Trigonometric Applications knowledge point
\[

$$\begin{aligned} &h = 790 \cdot \tan(27^\circ)\\ &\tan(27^\circ) \approx 0.509525\\ &h \approx 790 \cdot 0.509525 \approx 402.525\text{ ft}\\ &h \approx 402.5\text{ ft} \end{aligned}$$

\]
</reasoning>

<answer>
From a hot-air balloon, Violet measures a \(27^\circ\) angle of depression to a landmark that's 790 feet away, measuring horizontally. What's the balloon's vertical distance above the ground? Round your answer to the nearest tenth of a foot if necessary. <blank>402.5</blank> feet
</answer>

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Answer:

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<reasoning>

Identify the geometric relationship

Using the Trigonometric Applications and Right Triangle Trigonometry knowledge points
\[

$$\begin{aligned} &\theta = 27^\circ \quad \text{(angle of depression / elevation)}\\ &d = 790\text{ ft} \quad \text{(horizontal distance)}\\ &h = \text{vertical height above the ground} \end{aligned}$$

\]

Set up the trigonometric equation

Using the Right Triangle Trigonometry knowledge point
\[

$$\begin{aligned} &\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\\ &\tan(27^\circ) = \frac{h}{790} \end{aligned}$$

\]

Solve for the vertical distance

Using the Trigonometric Applications knowledge point
\[

$$\begin{aligned} &h = 790 \cdot \tan(27^\circ)\\ &\tan(27^\circ) \approx 0.509525\\ &h \approx 790 \cdot 0.509525 \approx 402.525\text{ ft}\\ &h \approx 402.5\text{ ft} \end{aligned}$$

\]
</reasoning>

<answer>
From a hot-air balloon, Violet measures a \(27^\circ\) angle of depression to a landmark that's 790 feet away, measuring horizontally. What's the balloon's vertical distance above the ground? Round your answer to the nearest tenth of a foot if necessary. <blank>402.5</blank> feet
</answer>

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