Question

question
from the observation deck of a skyscraper, sophia measures a 45° angle of depression to a ship in the harbor below.
if the observation deck is 870 feet high, what is the horizontal distance from the base of the skyscraper out to the
ship? round your answer to the nearest hundredth of a foot if necessary.
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Explanation:

Step1: Use the tangent function

In a right - triangle formed by the observation deck height, the horizontal distance, and the line of sight. The angle of depression is \(45^{\circ}\). The tangent of an angle in a right - triangle is given by \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 45^{\circ}\), the opposite side is the height of the observation deck (\(h = 870\) feet) and the adjacent side is the horizontal distance \(x\) (since \(\tan(45^{\circ})=\frac{870}{x}\)).
We know that \(\tan(45^{\circ}) = 1\). So, \(1=\frac{870}{x}\).

Step2: Solve for \(x\)

Cross - multiply the equation \(1=\frac{870}{x}\) to get \(x = 870\) feet.

Answer:

\(870\) feet