Question

from the observation deck of a skyscraper, meena measures a 48° angle of depression to a ship in the harbor below. if the observation deck is 1068 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: Set up the trigonometric relationship

The angle of depression is \(48^{\circ}\). The height of the observation deck (opposite side) is \(h = 1068\) feet. Let the horizontal distance (adjacent side) be \(x\). Since the angle of depression is equal to the angle of elevation from the ship to the observation deck, we use the tangent function: \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). So, \(\tan(48^{\circ})=\frac{1068}{x}\).

Step2: Solve for \(x\)

Rearrange the equation \(\tan(48^{\circ})=\frac{1068}{x}\) to get \(x=\frac{1068}{\tan(48^{\circ})}\).
We know that \(\tan(48^{\circ})\approx1.1106\). Then \(x = \frac{1068}{1.1106}\).

$$x=\frac{1068}{1.1106}\approx961.6$$

Answer:

\(961.6\) feet