A statue 1.6 m tall stands on the top of a pedestral . 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.

Answer

To find the height of the pedestal, we can use trigonometric relationships based on the angles of elevation given in the problem. Let's denote the following:

- Let \( H \) be the height of the pedestal.
- The height of the statue is given as \( 1.6 \) m.
- The total height from the ground to the top of the statue is \( H + 1.6 \) m.

### Step-by-Step Solution:

1. **Identify the angles of elevation:**
- The angle of elevation to the top of the statue is \( 60^\circ \).
- The angle of elevation to the top of the pedestal is \( 45^\circ \).

2. **Set up the equations using tangent:**
- From the point on the ground to the top of the statue:
\[ \tan(60^\circ) = \frac{H + 1.6}{d} \]
- From the point on the ground to the top of the pedestal:
\[ \tan(45^\circ) = \frac{H}{d} \]

Here, \( d \) is the horizontal distance from the point on the ground to the base of the pedestal.

3. **Use the values of tangent:**
- We know that \( \tan(60^\circ) = \sqrt{3} \) and \( \tan(45^\circ) = 1 \).
- Therefore, we can rewrite the equations:
\[ \sqrt{3} = \frac{H + 1.6}{d} \quad \text{(1)} \]
\[ 1 = \frac{H}{d} \quad \text{(2)} \]

4. **Express \( d \) in terms of \( H \):**
- From equation (2):
\[ d = H \quad \text{(3)} \]

5. **Substitute \( d \) from equation (3) into equation (1):**
- Substitute \( d \) in equation (1):
\[ \sqrt{3} = \frac{H + 1.6}{H} \]

6. **Cross-multiply to solve for \( H \):**
- Cross-multiplying gives:
\[ \sqrt{3}H = H + 1.6 \]
- Rearranging this gives:
\[ \sqrt{3}H - H = 1.6 \]
\[ H(\sqrt{3} - 1) = 1.6 \]

7. **Solve for \( H \):**
- Thus, we have:
\[ H = \frac{1.6}{\sqrt{3} - 1} \]

8. **Rationalize the denominator:**
- Multiply the numerator and denominator by \( \sqrt{3} + 1 \):
\[ H = \frac{1.6(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{1.6(\sqrt{3} + 1)}{3 - 1} = \frac{1.6(\sqrt{3} + 1)}{2} \]
- Simplifying gives:
\[ H = 0.8(\sqrt{3} + 1) \]

### Final Result:
The height of the pedestal is \( H = 0.8(\sqrt{3} + 1) \) meters.