To find the height of the building based on the angle of elevation and the distance from the base, we can use some basic 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 building).
The formula we can use is:
\[ \tan(\theta) = \frac{\text{height}}{\text{distance}} \]
where:
- \(\theta\) is the angle of elevation (19 degrees),
- height is the height of the building,
- distance is the distance from the base of the building (1 mile).
- Convert the distance from miles to feet. There are 5280 feet in a mile:
\[ \text{distance} = 1 \text{ mile} = 5280 \text{ feet} \]
- Rearranging the tangent function to solve for height gives us:
\[ \text{height} = \tan(\theta) \times \text{distance} \]
- Now, we can calculate the height:
\[ \text{height} = \tan(19^\circ) \times 5280 \]
- Using a calculator or trigonometric tables, we can find:
\(\tan(19^\circ) \approx 0.3443\).
- Now we can calculate the height:
\[ \text{height} = 0.3443 \times 5280 \approx 1817.544 \]
- Rounding to four decimal places, we get:
\[ \text{height} \approx 1817.5440 \text{ feet} \]
Thus, the height of the building is approximately 1817.5440 feet.