From a point A on the ground, the angle of elevation of the top of a 20 m tall buiding is `45^@`. A flag is hoisted at the top of the building and the angle of elevation of the top of the flagstaff from A is `60^@`. Find the length of the flag staff and the distance of the building from the point A.

Options

A. 20 m , 15 m
B. 20m, 14.64 m
C. 25 m , 15m
D. 20 m, 19m

Answer

To solve the problem, we will use trigonometric ratios, particularly the tangent function, which relates the angle of elevation to the opposite side (height) and the adjacent side (distance from the point).

### Step-by-Step Solution:

1. **Identify the elements of the problem:**
- Height of the building (h) = 20 m
- Angle of elevation to the top of the building (θ1) = 45°
- Angle of elevation to the top of the flagstaff (θ2) = 60°
- Let the distance from point A to the base of the building be d.

2. **Using the angle of elevation to the top of the building:**
- From the right triangle formed by point A, the base of the building, and the top of the building:
\[ \tan(θ1) = \frac{\text{height of building}}{\text{distance from A to building}} = \frac{h}{d} \]
- Plugging in the values:
\[ \tan(45°) = \frac{20}{d} \]
- Since \(\tan(45°) = 1\):
\[ 1 = \frac{20}{d} \implies d = 20 \text{ m} \]

3. **Finding the height of the flagstaff:**
- Let the height of the flagstaff be \(x\).
- The total height from point A to the top of the flagstaff is \(20 + x\).
- Using the angle of elevation to the top of the flagstaff:
\[ \tan(θ2) = \frac{\text{height of flagstaff + height of building}}{\text{distance from A to building}} = \frac{20 + x}{d} \]
- Plugging in the values:
\[ \tan(60°) = \frac{20 + x}{20} \]
- Since \(\tan(60°) = \sqrt{3}\):
\[ \sqrt{3} = \frac{20 + x}{20} \]
- Cross-multiplying gives:
\[ 20\sqrt{3} = 20 + x \]
- Solving for \(x\):
\[ x = 20\sqrt{3} - 20 \]

4. **Calculating the numerical value of the flagstaff height:**
- Approximating \(\sqrt{3} \approx 1.732\):
\[ x \approx 20(1.732) - 20 \approx 34.64 - 20 \approx 14.64 \text{ m} \]

### Final Results:
- **Length of the flagstaff**: \(x \approx 14.64 \text{ m}\)
- **Distance of the building from point A**: \(d = 20 \text{ m}\)