A 7 m long flagstaff is fixed on the top of a tower on the horizontal plane. From a point on the ground, the angles of elevation of the top and bottom of the flagstaff are `45^(@)" and " 30^(@)` respectively. Find the height of the tower.

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the problem
We have a flagstaff of height 7 m fixed on top of a tower. We need to find the height of the tower given the angles of elevation from a point on the ground to the top and bottom of the flagstaff are 45° and 30° respectively.

### Step 2: Draw a diagram
Let’s label the points:
- Let point A be the point on the ground from where the angles are measured.
- Let point B be the bottom of the flagstaff (top of the tower).
- Let point C be the top of the flagstaff.
- Let point D be the point directly below C on the ground.

### Step 3: Define the heights
Let the height of the tower be \( h \). Therefore, the height of the flagstaff above the tower is 7 m, making the total height from point A to point C equal to \( h + 7 \).

### Step 4: Use the tangent function for the angles of elevation
1. For angle of elevation to the top of the flagstaff (C):
\[ \tan(45^\circ) = \frac{AC}{AD} \]
Here, \( AC = h + 7 \) and \( AD = CD \) (the horizontal distance from point A to point D).

Since \( \tan(45^\circ) = 1 \):
\[ 1 = \frac{h + 7}{CD} \implies CD = h + 7 \]

2. For angle of elevation to the bottom of the flagstaff (B):
\[ \tan(30^\circ) = \frac{BC}{AD} \]
Here, \( BC = h \).

Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{h}{CD} \implies CD = h \sqrt{3} \]

### Step 5: Set the two expressions for CD equal to each other
From the two equations for \( CD \):
\[ h + 7 = h \sqrt{3} \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ h \sqrt{3} - h = 7 \implies h(\sqrt{3} - 1) = 7 \]

### Step 7: Solve for h
\[ h = \frac{7}{\sqrt{3} - 1} \]

### Step 8: Rationalize the denominator
To rationalize:
\[ h = \frac{7(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{7(\sqrt{3} + 1)}{3 - 1} = \frac{7(\sqrt{3} + 1)}{2} \]

### Step 9: Calculate the numerical value
Using \( \sqrt{3} \approx 1.732 \):
\[ h \approx \frac{7(1.732 + 1)}{2} = \frac{7(2.732)}{2} \approx \frac{19.124}{2} \approx 9.562 \text{ meters} \]

### Final Answer
The height of the tower is approximately \( 9.562 \) meters.

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