A vertical tower of height 20 m stands on a horizontal plane and is surmounted by a vertical flag staff of height h. At a point on the plane, the angle of elevation of the bottom and top of flag staff are `45^(@)` and `60^(@)`, respectively. Find the value of h.

Answer

To solve the problem, we will use trigonometric ratios and the information given about the angles of elevation.

### Step-by-step Solution:

1. **Understanding the Problem:**
- We have a vertical tower (AB) of height 20 m.
- A flagstaff (CD) of height \( h \) is on top of the tower.
- The angle of elevation to the bottom of the flagstaff (point C) is \( 45^\circ \).
- The angle of elevation to the top of the flagstaff (point D) is \( 60^\circ \).

2. **Setting Up the Diagram:**
- Let the distance from the point of observation (point A) to the base of the tower (point B) be \( x \).
- The height of the tower is \( AB = 20 \) m.
- The height of the flagstaff is \( CD = h \).
- The total height from point A to point D (top of the flagstaff) is \( AD = AB + CD = 20 + h \).

3. **Using Triangle ABC (for angle \( 45^\circ \)):**
- In triangle ABC, we can use the tangent function:
\[ \tan(45^\circ) = \frac{AB}{x} = \frac{20}{x} \]
- Since \( \tan(45^\circ) = 1 \):
\[ 1 = \frac{20}{x} \implies x = 20 \text{ m} \]

4. **Using Triangle ABD (for angle \( 60^\circ \)):**
- In triangle ABD, we again use the tangent function:
\[ \tan(60^\circ) = \frac{AD}{x} = \frac{20 + h}{x} \]
- Since \( \tan(60^\circ) = \sqrt{3} \):
\[ \sqrt{3} = \frac{20 + h}{20} \]
- Cross-multiplying gives:
\[ 20\sqrt{3} = 20 + h \]
- Rearranging for \( h \):
\[ h = 20\sqrt{3} - 20 \]

5. **Calculating the Value of \( h \):**
- We can factor out 20:
\[ h = 20(\sqrt{3} - 1) \]
- Using the approximate value \( \sqrt{3} \approx 1.732 \):
\[ h \approx 20(1.732 - 1) = 20(0.732) \approx 14.64 \text{ m} \]

### Final Answer:
The height of the flagstaff \( h \) is approximately \( 14.64 \) meters.

---