Question

from a hot - air balloon, ian measures a 30° angle of depression to a landmark thats 1250 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:

Step1: Identify the trigonometric relationship

We have a right triangle where the horizontal distance (adjacent side) is 1250 feet, the vertical distance (opposite side, let's call it \( h \)) is what we need to find, and the angle of depression is \( 30^\circ \). The angle of depression is equal to the angle of elevation from the landmark to the balloon, so we can use the tangent function: \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \). Here, \( \theta = 30^\circ \), opposite side is \( h \), adjacent side is 1250.

Step2: Set up the equation and solve

Using \( \tan(30^\circ)=\frac{h}{1250} \). We know that \( \tan(30^\circ)=\frac{1}{\sqrt{3}}\approx0.57735 \). So, \( h = 1250\times\tan(30^\circ) \).
Calculating \( 1250\times\frac{1}{\sqrt{3}}=\frac{1250}{\sqrt{3}}\approx\frac{1250}{1.73205}\approx721.7 \) (rounded to the nearest tenth).

Answer:

The balloon's vertical distance above the ground is approximately 721.7 feet.