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

In a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). The angle of depression is \(67^{\circ}\), and the height (opposite side) is \(1183\) feet, and the horizontal distance \(x\) is the adjacent side. Since the angle of depression is equal to the angle of elevation in this case, \(\tan(67^{\circ})=\frac{1183}{x}\).

Step2: Solve for \(x\)

We can rewrite the equation as \(x = \frac{1183}{\tan(67^{\circ})}\). We know that \(\tan(67^{\circ})\approx2.3559\). Then \(x=\frac{1183}{2.3559}\).

Step3: Calculate the value of \(x\)

\(x=\frac{1183}{2.3559}\approx502.1\)

Answer:

\(502.1\) feet