The respective ratio between the height of tower and the point at some distance from its foot is 57 : `19sqrt3` What is the angle (in degrees) of elevation of the top of the tower?

Options

A. 30
B. 45
C. 60
D. 75

Answer

To find the angle of elevation of the top of the tower, we can use the given ratio of the height of the tower to the distance from its foot. Let's denote:

- Height of the tower = \( h \)
- Distance from the foot of the tower = \( d \)

According to the problem, the ratio of the height of the tower to the distance from its foot is given as:

\[ \frac{h}{d} = \frac{57}{19\sqrt{3}} \]

From this ratio, we can express \( h \) in terms of \( d \):

\[ h = \frac{57}{19\sqrt{3}} \cdot d \]

Next, we will use the tangent of the angle of elevation \( \theta \) to find the angle. The tangent of the angle of elevation is given by:

\[ \tan(\theta) = \frac{\text{Height of the tower}}{\text{Distance from the foot}} = \frac{h}{d} \]

Substituting the expression for \( h \):

\[ \tan(\theta) = \frac{57}{19\sqrt{3}} \]

Now, we need to find the angle \( \theta \) such that:

\[ \tan(\theta) = \frac{57}{19\sqrt{3}} \]

To find \( \theta \), we can use the inverse tangent function:

\[ \theta = \tan^{-1}\left(\frac{57}{19\sqrt{3}}\right) \]

Now, we can calculate \( \frac{57}{19\sqrt{3}} \):

1. Calculate \( 19\sqrt{3} \):
- \( \sqrt{3} \approx 1.732 \)
- \( 19\sqrt{3} \approx 19 \times 1.732 \approx 32.888 \)

2. Now, calculate \( \frac{57}{32.888} \):
- \( \frac{57}{32.888} \approx 1.733 \)

Now, we find \( \theta \):

Using a calculator, we find:

\[ \theta \approx \tan^{-1}(1.733) \approx 60^\circ \]

Thus, the angle of elevation of the top of the tower is:

\[ \theta \approx 60^\circ \]

### Summary of Steps:
1. Set up the ratio of height to distance.
2. Express height in terms of distance using the ratio.
3. Use the tangent function to relate height and distance to the angle of elevation.
4. Calculate the value of the tangent.
5. Use the inverse tangent function to find the angle.