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?

## Step 1: Identify the right triangle<br />### 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.<br />## Step 2: Use trigonometric function<br />### 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).<br />\[<br />\tan(30^\circ) = \frac{\text{height}}{767.6}<br />\]<br />## Step 3: Solve for the height<br />### Rearrange the equation to solve for the height:<br />\[<br />\text{height} = 767.6 \times \tan(30^\circ)<br />\]<br />### Calculate the value using the known tangent of 30 degrees (\(\tan(30^\circ) = \frac{1}{\sqrt{3}}\)):<br />\[<br />\text{height} = 767.6 \times \frac{1}{\sqrt{3}}<br />\]<br />\[<br />\text{height} \approx 767.6 \times 0.5774<br />\]<br />\[<br />\text{height} \approx 443.5 \text{ meters}<br />\]