Question

from the observation deck of a skyscraper, adrian measures a 67° angle of depression to a ship in the harbor below. if the observation deck is 1183 feet high, what is the horizontal distance from the base of the skyscraper out to the ship? round your answer to the nearest tenth of a foot if necessary.

Explanation:

Step1: Use the tangent function

The angle of depression is \(67^{\circ}\). The vertical side (height of the observation deck) is \(1183\) feet, and the horizontal side is \(x\). Since the angle of depression is equal to the angle of elevation in this right - triangle situation, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 67^{\circ}\), the opposite side is \(1183\) and the adjacent side is \(x\). So, \(\tan(67^{\circ})=\frac{1183}{x}\).

Step2: Solve for \(x\)

We can rewrite the equation as \(x=\frac{1183}{\tan(67^{\circ})}\). We know that \(\tan(67^{\circ})\approx2.355852366\). Then \(x=\frac{1183}{2.355852366}\).

Answer:

\(x\approx502.2\) feet