Question

( Trig. Functions and Angles of Elevation & Depression)

A spotlight (which is mounted at ground level) is pointing to the top of the Empire State Building with an angle of elevation of 78°
. If the horizontal (along the ground) distance between the building and the spotlight is is 309 feet, how tall is the Empire State Building?

Answer 1

To solve this problem, we can use trigonometric functions and the concept of angles of elevation and depression.

Let's first define the terms "angle of elevation" and "angle of depression". An angle of elevation is the angle between a horizontal line and a line from the observer's eye to an object above the horizontal line. An angle of depression is the angle between a horizontal line and a line from the observer's eye to an object below the horizontal line.

In this problem, we are given that the angle of elevation from the spotlight to the top of the Empire State Building is 78°. We can use this information to find the height of the building.

X represents the point where the spotlight hits the Empire State Building. We want to find the height of the building, which we can call h.

We know that tan(78°) = h/309. To solve for h, we can multiply both sides by 309:

h = 309 * tan(78°)

Using a calculator, we get:

h ≈ 1289.5 feet

Therefore, the Empire State Building is approximately 1,289.5 feet tall.