Question

from the observation deck of a skyscraper, aaliyah measures a 67° angle of depression to a ship in the harbor below. if the observation deck is 883 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 \(67^{\circ}\), and the height of the observation deck (opposite side) is \(883\) feet. Let \(x\) be the horizontal distance (adjacent side). The tangent of an angle in a right - triangle is \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Since the angle of depression is equal to the angle of elevation from the ship to the observation deck, \(\tan(67^{\circ})=\frac{883}{x}\).

Step2: Solve for \(x\)

We can rewrite the equation as \(x = \frac{883}{\tan(67^{\circ})}\). We know that \(\tan(67^{\circ})\approx2.3559\). Then \(x=\frac{883}{2.3559}\).

Step3: Calculate the value of \(x\)

\(x=\frac{883}{2.3559}\approx375.7\)

Answer:

\(375.7\) feet