If an observer is 200 ft from a building and the angle of elevationto the top of a building is 60°,what is the height of the building?

recall that the sides of a 30-60-90 triangle are in the ratio 1:√3:2

You have a side of 200ft opposite the 30° angle, so it is the shortest side.

To find the height of the building, we can use the tangent function. The tangent of an angle is equal to the opposite side divided by the adjacent side.

In this case, the opposite side is the height of the building and the adjacent side is the distance from the observer to the building.

We can set up the equation as follows:

tan(60°) = (height of building) / 200 ft

To solve for the height of the building, we can rearrange the equation:

(height of building) = tan(60°) * 200 ft

Now let's calculate the height of the building:

Using a calculator, we can find that tan(60°) is approximately 1.732.

(height of building) = 1.732 * 200 ft

Therefore, the height of the building is approximately 346.4 ft.

To find the height of the building, we can use trigonometry. In this case, we need to use the tangent function. Here's how you can solve the problem step-by-step:

1. Draw a diagram: Sketch a right triangle where the vertical side represents the height of the building, the horizontal side represents the distance from the observer to the building, and the angle of elevation is 60°.

2. Identify the known values: In this case, the distance from the observer to the building is 200 ft, and the angle of elevation is 60°.

3. Set up the trigonometric equation: The tangent of an angle is defined as the ratio of the opposite side to the adjacent side of a right triangle. In this case, the opposite side is the height of the building, and the adjacent side is the distance from the observer to the building. Therefore, we have tan(60°) = height/200.

4. Solve the equation: Using a scientific calculator or a trigonometric table, find the tangent of 60°, which is √3. The equation becomes √3 = height/200.

5. Rearrange the equation to solve for the height: Multiply both sides of the equation by 200 to isolate the height. The equation becomes height = 200 * √3.

6. Calculate the height: Multiply 200 by the square root of 3 (√3) to find the height. The approximate value is height ≈ 346.41 ft.

Therefore, the height of the building is approximately 346.41 ft.