Question

from a hot-air balloon, genesis measures a $40^\circ$ angle of depression to a landmark thats 799 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:

Step1: Define trigonometric relationship

The angle of depression equals the angle of elevation from the landmark to the balloon. We use the tangent function: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$, where $\theta=40^\circ$, opposite side is $x$ (vertical distance), adjacent side is 799 feet.

Step2: Rearrange to solve for $x$

Isolate $x$ by multiplying both sides by 799.
$x = 799 \times \tan(40^\circ)$

Step3: Calculate the value

Use $\tan(40^\circ) \approx 0.8391$.
$x \approx 799 \times 0.8391$
$x \approx 670.44$

Answer:

670.44 feet