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 hoisted 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 `sqrt(3)=1.732)`

Options

A. `10(sqrt(3)+2)`m
B. `10(sqrt(3)+1)`m
C. `10sqrt(3)` m
D. 7.32 m

Answer

To solve the problem step by step, we will use trigonometric principles.

### Step 1: Understand the scenario
We have a point P on the ground, a building of height 10 meters, and a flagstaff on top of the building. The angle of elevation to the top of the building from point P is \(30^\circ\), and the angle of elevation to the top of the flagstaff is \(45^\circ\).

### Step 2: Set up the triangle for the building
Let the distance from point P to the base of the building be \(d\). According to the tangent function:
\[ \tan(30^\circ) = \frac{\text{height of the building}}{\text{distance from P to the building}} = \frac{10}{d} \]
Using the value of \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\):
\[ \frac{1}{\sqrt{3}} = \frac{10}{d} \]
Cross-multiplying gives:
\[ d = 10\sqrt{3} \]

### Step 3: Set up the triangle for the flagstaff
Let the height of the flagstaff be \(x\). The total height from the ground to the top of the flagstaff is \(10 + x\). The angle of elevation to the top of the flagstaff is \(45^\circ\):
\[ \tan(45^\circ) = \frac{\text{total height}}{\text{distance from P to the building}} = \frac{10 + x}{d} \]
Since \(\tan(45^\circ) = 1\):
\[ 1 = \frac{10 + x}{d} \]
Substituting \(d\) from the previous step:
\[ 1 = \frac{10 + x}{10\sqrt{3}} \]
Cross-multiplying gives:
\[ 10\sqrt{3} = 10 + x \]

### Step 4: Solve for \(x\)
Rearranging the equation:
\[ x = 10\sqrt{3} - 10 \]

### Step 5: Substitute \(\sqrt{3}\)
Using the given value \(\sqrt{3} = 1.732\):
\[ x = 10(1.732) - 10 \]
Calculating:
\[ x = 17.32 - 10 = 7.32 \]

### Conclusion
The length of the flagstaff is \(7.32\) meters.