Question

1. the angle of elevation to the top of a building in houston is found to be 9° from the ground at a distance of 1 km from the base of the building. find the height of the building.
2. if a building is 200 m tall and you are 400 m from the building, what is the angle of elevation from where you are standing to the top of the building?
3. a radio tower is located 325 feet from a building. from a window in the building, a person determines that the angle of elevation to the top of the tower is 43° and that the angle of depression to the bottom of the tower is 31°. how tall is the tower?
4. a 400 foot tall monument is located in the distance. from a window in a building, a person determines that the angle of elevation to the top of the monument is 18° and that the angle of depression to the bottom of the tower is 3°. how far is the person from the monument?

Explanation:

Step1: Convert units and recall trigonometric formula

We know that \(1\ km = 1000\ m\). For a right - triangle, if the angle of elevation is \(\theta\), the distance from the base of the object is \(x\), and the height of the object is \(h\), then \(\tan\theta=\frac{h}{x}\). Here, \(\theta = 9^{\circ}\), \(x = 1000\ m\), and we need to find \(h\).

Step2: Calculate the height

Using the formula \(h=x\tan\theta\), substitute \(x = 1000\) and \(\theta=9^{\circ}\). We know that \(\tan(9^{\circ})\approx0.1584\). Then \(h = 1000\times\tan(9^{\circ})\).

$$h=1000\times0.1584 = 158.4\ m$$

Answer:

The height of the building is approximately \(158.4\ m\)