Understanding the Problem
To determine the new angle of elevation when the height of the tower is tripled, we first need to understand the relationship between the height of the tower and the angle of elevation from a given point on the ground.
Initial Setup
- Let the height of the tower be \( h \).
- The distance from the point on the ground to the base of the tower is \( d \).
- The initial angle of elevation is given as \( 30^\circ \).
Using the tangent function in trigonometry:
\[
\tan(30^\circ) = \frac{h}{d}
\]
Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), we can express the relationship as:
\[
\frac{h}{d} = \frac{1}{\sqrt{3}} \implies h = \frac{d}{\sqrt{3}}
\]
Tripling the Height
Now, if the height of the tower is tripled, the new height becomes:
\[
h' = 3h = 3\left(\frac{d}{\sqrt{3}}\right) = \frac{3d}{\sqrt{3}} = \sqrt{3}d
\]
Finding the New Angle of Elevation
Now we calculate the new angle of elevation, \( \theta \):
\[
\tan(\theta) = \frac{h'}{d} = \frac{\sqrt{3}d}{d} = \sqrt{3}
\]
Since \(\tan(60^\circ) = \sqrt{3}\), we find that:
\[
\theta = 60^\circ
\]
Conclusion
- The angle of elevation, when the height of the tower is tripled, becomes \( 60^\circ \).
- This demonstrates how the angle of elevation increases with an increase in tower height, given a constant distance from the base.