The elevation of a tower at a station A due north of it is `45^@` and at a station B due west of A is `30^@`.
If AB=40 m, find the height of the tower:

Options

A. 26.26m
B. 28.28m
C. 38.5m
D. none of these

Answer

To find the height of the tower based on the given angles of elevation from points A and B, we can follow these steps:

### Step 1: Understand the Geometry
We have two points A and B:
- Point A is due north of the tower with an angle of elevation of \(45^\circ\).
- Point B is due west of A with an angle of elevation of \(30^\circ\).
- The distance \(AB\) is given as 40 m.

### Step 2: Set Up the Diagram
Let:
- \(H\) be the height of the tower.
- \(D_A\) be the distance from point A to the base of the tower.
- \(D_B\) be the distance from point B to the base of the tower.

From the problem, we can visualize the scenario as a right triangle for both points A and B.

### Step 3: Use Trigonometric Ratios
From point A:
\[ \tan(45^\circ) = \frac{H}{D_A} \]
Since \(\tan(45^\circ) = 1\):
\[ H = D_A \quad \text{(1)} \]

From point B:
\[ \tan(30^\circ) = \frac{H}{D_B} \]
Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\):
\[ H = \frac{D_B}{\sqrt{3}} \quad \text{(2)} \]

### Step 4: Relate Distances
The distance \(AB\) can be expressed in terms of \(D_A\) and \(D_B\):
\[ AB = D_A + D_B = 40 \quad \text{(3)} \]

### Step 5: Substitute Equations
From equation (1), we can substitute \(D_A\) in terms of \(H\) into equation (3):
\[ D_A + D_B = 40 \]
Substituting \(D_A = H\) into the equation gives:
\[ H + D_B = 40 \quad \text{(4)} \]

Now, substitute \(D_B\) from equation (2):
\[ H + \sqrt{3}H = 40 \]
This simplifies to:
\[ H(1 + \sqrt{3}) = 40 \]

### Step 6: Solve for \(H\)
Now, we can solve for \(H\):
\[ H = \frac{40}{1 + \sqrt{3}} \]

### Step 7: Rationalize the Denominator
To simplify \(H\), we can multiply the numerator and denominator by the conjugate of the denominator:
\[ H = \frac{40(1 - \sqrt{3})}{(1 + \sqrt{3})(1 - \sqrt{3})} = \frac{40(1 - \sqrt{3})}{1 - 3} = \frac{40(1 - \sqrt{3})}{-2} = -20(1 - \sqrt{3}) \]
Thus:
\[ H = 20(\sqrt{3} - 1) \]

### Step 8: Calculate the Height
Using an approximate value for \(\sqrt{3} \approx 1.732\):
\[ H \approx 20(1.732 - 1) \approx 20(0.732) \approx 14.64 \text{ m} \]

Therefore, the height of the tower is approximately \(14.64\) meters.

---