Question

from the observation deck of a skyscraper, sophia measures a $45^{circ}$ angle of depression to a ship in the harbor below. if the observation deck is 870 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 triangle type

The angle of depression is \(45^\circ\), so the right triangle formed has a \(45^\circ\) angle, making it a 45 - 45 - 90 triangle (isosceles right triangle), where the legs are equal.

Step2: Relate height and horizontal distance

In a 45 - 45 - 90 triangle, the two legs (height of the deck and horizontal distance \(x\)) are equal. The height of the observation deck is 870 feet, so the horizontal distance \(x\) is also 870 feet. Mathematically, using the tangent function: \(\tan(45^\circ)=\frac{\text{opposite}}{\text{adjacent}}=\frac{870}{x}\). Since \(\tan(45^\circ) = 1\), we have \(1=\frac{870}{x}\), which gives \(x = 870\) when we solve for \(x\) (multiplying both sides by \(x\) and then dividing by 1).

Answer:

870.00