Question

from a hot - air balloon, logan measures a 40° angle of depression to a landmark thats 1892 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: Use the tangent function

The angle of depression is \(40^{\circ}\). In the right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 40^{\circ}\), and the adjacent side to the angle (in the right - triangle) is \(x = 1892\) feet. Let the vertical distance be \(h\). Since the angle of depression is equal to the angle of elevation in the right - triangle (alternate interior angles), \(\tan40^{\circ}=\frac{h}{1892}\).

Step2: Solve for \(h\)

We know that \(\tan40^{\circ}\approx0.8391\). Then \(h = 1892\times\tan40^{\circ}\). Substitute the value of \(\tan40^{\circ}\): \(h=1892\times0.8391\).

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Answer:

\(1587.6\) feet