Question

question
from the observation deck of a skyscraper, jack measures a ( 48^circ ) angle of depression to a ship in the harbor below. if the observation deck is 1181 feet high, what is the horizontal distance from the base of the skyscraper out to the ship? round your answer to the nearest hundredth of a foot if necessary.

Explanation:

Step1: Identify the trigonometric relationship

We have a right triangle where the vertical side (height of the observation deck) is 1181 feet, the horizontal side is the distance \( x \) we need to find, and the angle of depression is \( 48^\circ \). The angle of depression is equal to the angle of elevation from the ship to the observation deck, so we can use the tangent function. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. So, \( \tan(48^\circ)=\frac{\text{opposite}}{\text{adjacent}}=\frac{1181}{x} \).

Step2: Solve for \( x \)

We can rearrange the formula to solve for \( x \). From \( \tan(48^\circ)=\frac{1181}{x} \), we get \( x = \frac{1181}{\tan(48^\circ)} \). Now we calculate \( \tan(48^\circ)\approx1.1106 \). Then \( x=\frac{1181}{1.1106}\approx1063.39 \).

Answer:

\( 1063.39 \)