To determine the height of the skyscraper, we can use the information provided and apply some basic trigonometry.
-
Identify the given information:
- Height from Hannah's eye level to the ground: \( h_{Hannah} = 1.68 \) meters
- Angle of elevation to the top of the skyscraper: \( \theta = 31^\circ \)
- Horizontal distance from Hannah to the base of the skyscraper: \( d = 194 \) meters
-
Determine the height of the skyscraper above Hannah's eye level:
- The relationship between the height of the skyscraper above her eye level and the angle of elevation and distance can be established using the tangent function: \[ \tan(\theta) = \frac{h_{skyscraper} - h_{Hannah}}{d} \] Rearranging this equation gives: \[ h_{skyscraper} - h_{Hannah} = d \cdot \tan(\theta) \] Therefore: \[ h_{skyscraper} = h_{Hannah} + d \cdot \tan(\theta) \]
-
Calculate the height of the skyscraper: \[ h_{skyscraper} = 1.68 + 194 \cdot \tan(31^\circ) \]
-
Calculate \(\tan(31^\circ)\): Using a calculator: \[ \tan(31^\circ) \approx 0.6009 \]
-
Substitute this value into the equation: \[ h_{skyscraper} = 1.68 + 194 \cdot 0.6009 \] \[ h_{skyscraper} = 1.68 + 116.5196 \] \[ h_{skyscraper} \approx 118.1996 \text{ meters} \]
-
Round the final height to the nearest hundredth: \[ h_{skyscraper} \approx 118.20 \text{ meters} \]
Therefore, the height of the skyscraper is approximately 118.20 meters.