A 10 m long flagstaff is fixed on the top of a tower from a point on the ground, the angles of elevations of the top and bottom of flagstaff are `45^(@) and 30^(@)` respectively. Find the height of the tower

Answer

To solve the problem, we need to find the height of the tower given the angles of elevation to the top and bottom of a flagstaff. Here’s a step-by-step solution:

### Step 1: Understand the Geometry
We have a tower (BC) and a flagstaff (AB) on top of it. The height of the flagstaff (AB) is given as 10 m. The angles of elevation from a point on the ground to the bottom (A) and top (B) of the flagstaff are 30° and 45°, respectively.

### Step 2: Set Up the Problem
Let:
- \( h \) = height of the tower (BC)
- The total height from the ground to the top of the flagstaff (B) = \( h + 10 \) m

### Step 3: Use the Angle of Elevation to Set Up Equations
1. From the point on the ground to the top of the flagstaff (B), the angle of elevation is 45°:
\[ \tan(45°) = \frac{h + 10}{x} \]
Since \( \tan(45°) = 1 \):
\[ 1 = \frac{h + 10}{x} \implies x = h + 10 \quad \text{(Equation 1)} \]

2. From the point on the ground to the bottom of the flagstaff (A), the angle of elevation is 30°:
\[ \tan(30°) = \frac{h}{x} \]
Since \( \tan(30°) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{h}{x} \implies x = \sqrt{3}h \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 2 into Equation 1
From Equation 1, we have:
\[ x = h + 10 \]
From Equation 2, we have:
\[ x = \sqrt{3}h \]
Setting these two expressions for \( x \) equal to each other:
\[ h + 10 = \sqrt{3}h \]

### Step 5: Solve for \( h \)
Rearranging the equation:
\[ \sqrt{3}h - h = 10 \]
Factoring out \( h \):
\[ h(\sqrt{3} - 1) = 10 \]
Now, solve for \( h \):
\[ h = \frac{10}{\sqrt{3} - 1} \]

### Step 6: Rationalize the Denominator
To simplify \( h \):
\[ h = \frac{10(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{10(\sqrt{3} + 1)}{3 - 1} = \frac{10(\sqrt{3} + 1)}{2} \]
\[ h = 5(\sqrt{3} + 1) \]

### Step 7: Calculate the Numerical Value
Using \( \sqrt{3} \approx 1.732 \):
\[ h \approx 5(1.732 + 1) = 5(2.732) \approx 13.66 \text{ m} \]

### Conclusion
The height of the tower is approximately \( 13.66 \) meters.

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