From a point P on the ground the angle of elevation of the top of a 10 m tall building is $$30^∘$$​. A flag is heisted at the top of the building and the angle of elevation of the top of the flagstaff from P is $$45^∘$$​. Find the length of the flagstaff. (Take √3=1.732 )

Correct option is D

Solution

We are tasked to find the length of the flagstaff given:

- The height of the building: $$10 {m}$$​

- Angle of elevation to the top of the building: $$30^\circ$$​

- Angle of elevation to the top of the flagstaff: $$45^\circ$$​

Step-by-Step Solution:

1. Given Data:

- Let the distance between the point P and the base of the building be d m.

- Let the height of the flagstaff be $$h_f \,{m}$$​ .

- The total height of the building and flagstaff is H = 10 + $$h_f$$​ .

2. Relation for the Building:

Using the tangent function for the angle of elevation to the top of the building $$( 30^\circ )$$​:

$$\tan 30^\circ = \frac{\text{Height of the building}}{\text{Distance from the building}}$$​

Substituting values:

$$\tan 30^\circ = \frac{10}{d}, \quad \tan 30^\circ = \frac{1}{\sqrt{3}} = \frac{1}{1.732}$$​

Rearranging:

$$d = 10 \times 1.732 = 17.32 \, \text{m}.$$​

3. Relation for the Building and Flagstaff:

Using the tangent function for the angle of elevation to the top of the flagstaff $$( 45^\circ )$$​:

$$\tan 45^\circ = \frac{\text{Total Height (Building + Flagstaff)}}{\text{Distance from the building}}$$​

Substituting values:

$$\tan 45^\circ = 1, \quad 1 = \frac{10 + h_f}{d}$$​

Substituting d = 17.32 :

$$1 = \frac{10 + h_f}{17.32}$$​

Rearranging:

$$10 + h_f = 17.32$$​

4. Solve for $$h_f$$​ :

Subtract 10 from both sides:

$$h_f = 17.32 - 10 = 7.32 \, \text{m}$$​.

Final Answer:

$$\boxed{7.32 \, \text{m}}$$​