A statue 1.46 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is `60^(@)` and from the same point the angle of elevation of the top of the pedestal is `45^(@)`. Find the height of the pedestal (use `sqrt(3)` = 1.73).

Answer

To find the height of the pedestal, we can follow these steps:

1. **Identify the given information:**
- Height of the statue (AB) = 1.46 m
- Angle of elevation to the top of the statue (A) = 60°
- Angle of elevation to the top of the pedestal (C) = 45°

2. **Set up the problem using trigonometric ratios:**
- Let the height of the pedestal be \( H \).
- The total height from the ground to the top of the statue (D) is \( H + 1.46 \) m.
- Let the distance from the point on the ground to the base of the pedestal (point B) be \( X \).

3. **Use the tangent function for the angle of elevation to the top of the pedestal:**
- From triangle ABC (where C is the top of the pedestal):
\[ \tan(45^\circ) = \frac{H}{X} \]
Since \( \tan(45^\circ) = 1 \):
\[ H = X \quad (1) \]

4. **Use the tangent function for the angle of elevation to the top of the statue:**
- From triangle ABD (where D is the top of the statue):
\[ \tan(60^\circ) = \frac{H + 1.46}{X} \]
Since \( \tan(60^\circ) = \sqrt{3} \):
\[ \sqrt{3} = \frac{H + 1.46}{X} \]
Rearranging gives:
\[ H + 1.46 = \sqrt{3} \cdot X \quad (2) \]

5. **Substitute equation (1) into equation (2):**
- Replace \( X \) with \( H \):
\[ H + 1.46 = \sqrt{3} \cdot H \]
Rearranging gives:
\[ 1.46 = \sqrt{3} \cdot H - H \]
\[ 1.46 = H(\sqrt{3} - 1) \]

6. **Solve for \( H \):**
\[ H = \frac{1.46}{\sqrt{3} - 1} \]
Now substituting \( \sqrt{3} \approx 1.73 \):
\[ H = \frac{1.46}{1.73 - 1} = \frac{1.46}{0.73} \]
\[ H \approx 2.00 \text{ m} \]

Thus, the height of the pedestal is approximately 2 meters.