Questions: A lamppost is located 152 yards from a building. The angle of elevation from the base of the lamppost to the top of the building is 33.4 degrees. Approximately how tall is the building?

Solution Steps

To find the height of the building, we can use trigonometry. Specifically, we can use the tangent function, which relates the angle of elevation to the opposite side (height of the building) and the adjacent side (distance from the lamppost to the building). The formula is:

\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]

Rearranging to solve for the height (opposite side):

\[ \text{height} = \tan(\theta) \times \text{adjacent} \]

Where:

• \(\theta\) is the angle of elevation (33.4 degrees)
• The adjacent side is the distance from the lamppost to the building (152 yards)

We are given:

• Distance from the lamppost to the building: \( d = 152 \) yards
• Angle of elevation to the top of the building: \( \theta = 33.4^\circ \)

To use the tangent function, we convert the angle from degrees to radians: \[ \theta_{\text{radians}} = \frac{33.4 \times \pi}{180} \approx 0.5829 \]

Using the tangent function, we can express the height \( h \) of the building as: \[ \tan(\theta) = \frac{h}{d} \] Rearranging gives: \[ h = d \cdot \tan(\theta) \]

Substituting the known values: \[ h = 152 \cdot \tan(0.5829) \approx 100.23 \text{ yards} \]

Final Answer