Question

question
from a hot - air balloon, santiago measures a 21° angle of depression to a landmark thats 569 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: Use trigonometric ratio

Since the angle of depression is \(21^{\circ}\), the angle inside the right - triangle (opposite to the side we want to find) is also \(21^{\circ}\) (alternate interior angles). We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 21^{\circ}\) and the adjacent side \(x\) (horizontal distance) is related to the vertical side (height \(h\)) in the right - triangle. But we can also use \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\) where the opposite side is the vertical distance we want to find and the adjacent side is \(569\) (the horizontal distance from the landmark to the point directly below the balloon).

We know that \(\tan(21^{\circ})=\frac{h}{569}\), so \(h = 569\times\tan(21^{\circ})\)

Step2: Calculate the value

We know that \(\tan(21^{\circ})\approx0.3839\)

Then \(h = 569\times0.3839\)

\(h=569\times0.3839=(500 + 60+9)\times0.3839=(500\times0.3839)+(60\times0.3839)+(9\times0.3839)\)

\(500\times0.3839 = 191.95\), \(60\times0.3839=23.034\), \(9\times0.3839 = 3.4551\)

\(h=191.95+23.034 + 3.4551=218.4391\approx218.44\)

Answer:

\(218.44\)