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Luis Is Standing on a Street in New York City Looking at the Top of the Empire State Building with a 30^circ Angle of Elevation. He Is

Question

Luis is standing on a street in New York City looking at the top of the Empire State Building with a 30^circ angle of elevation. He is 767.6 meters from the base of the Empire State Building How tall is the Empire State Building?

Solution

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Answer

### The height of the Empire State Building is approximately 443.5 meters.

Explanation

## Step 1: Identify the right triangle### Luis, the base of the Empire State Building, and the top of the Empire State Building form a right triangle. The distance from Luis to the base is one leg, and the height of the building is the other leg.## Step 2: Use trigonometric function### The tangent of the angle of elevation (30 degrees) relates the opposite side (height of the building) to the adjacent side (distance from Luis to the base). ## Step 3: Solve for the height### Rearrange the equation to solve for the height: ### Calculate the value using the known tangent of 30 degrees (\(\tan(30^\circ) = \frac{1}{\sqrt{3}}\)):