Question

from the observation deck of a skyscraper, lavaughn measures a 42° angle of depression to a ship in the harbor below. if the observation deck is 872 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 right - triangle relationship

The angle of depression is equal to the angle of elevation from the ship to the observation deck. Let the horizontal distance be $x$. We know the height of the observation deck (opposite side) is 872 feet and the angle of elevation is 42°. We use the tangent function $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.
$\tan(42^{\circ})=\frac{872}{x}$

Step2: Solve for $x$

We can rewrite the equation as $x = \frac{872}{\tan(42^{\circ})}$.
Since $\tan(42^{\circ})\approx0.9004$, then $x=\frac{872}{0.9004}\approx968.46$ feet.

Answer:

968.46 feet