Question

from the observation deck of a skyscraper, rahul measures a 48° angle of depression to a ship in the harbor below. if the observation deck is 1093 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: Identify the trigonometric relationship

We know that the angle of depression is \(48^{\circ}\). The height of the observation deck (opposite side) is \(h = 1093\) feet and we need to find the horizontal distance \(x\) (adjacent side). Since the angle of depression is equal to the angle of elevation in this right - triangle situation, we use the tangent function: \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 48^{\circ}\), opposite \(=1093\) and adjacent \(=x\). So, \(\tan(48^{\circ})=\frac{1093}{x}\).

Step2: Solve for \(x\)

We can rewrite the equation from Step 1 as \(x=\frac{1093}{\tan(48^{\circ})}\).
We know that \(\tan(48^{\circ})\approx1.1106\). Then \(x=\frac{1093}{1.1106}\).

$$x=\frac{1093}{1.1106}\approx984.2$$

Answer:

The horizontal distance from the base of the skyscraper out to the ship is approximately \(984.2\) feet.