Question

from a hot-air balloon, isaiah measures a $22^\circ$ angle of depression to a landmark thats 312 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: Identify trigonometric ratio

The angle of depression equals the angle of elevation from the landmark to the balloon ($22^\circ$). We use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$, where $\theta=22^\circ$, adjacent side = 312 ft, opposite side = $x$ (vertical distance).

Step2: Set up the equation

$\tan(22^\circ) = \frac{x}{312}$

Step3: Solve for $x$

Rearrange to isolate $x$: $x = 312 \times \tan(22^\circ)$
Calculate $\tan(22^\circ) \approx 0.4040$, so $x \approx 312 \times 0.4040$

Step4: Compute the value

$x \approx 126.048$

Answer:

126.05 feet