Question

from the observation deck of a skyscraper, isabella measures a 45° angle of depression to a ship in the harbor below. if the observation deck is 1036 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: Use the tangent function

The angle of depression is \(45^{\circ}\). The tangent of an angle in a right - triangle is given by \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\theta = 45^{\circ}\), the opposite side is the height of the observation deck (\(h = 1036\) feet), and the adjacent side is the horizontal distance \(x\) (which we want to find). So, \(\tan(45^{\circ})=\frac{1036}{x}\).

Step2: Solve for \(x\)

Since \(\tan(45^{\circ}) = 1\), the equation \(1=\frac{1036}{x}\) implies \(x = 1036\) (by cross - multiplying: \(x\times1=1036\)).

Answer:

\(1036.00\)