If the angle of elevation of a tower from two distant points a and b (a > b) from its foot and in the same straight line and on the same side of it are `30^@ and 60^@`, then the height of the tower is

Options

A. a )`sqrt(a/b)`
B. `b ) sqrt(a+b)`
C. c) `sqrt(ab)`
D. d ) `sqrt(a-b)`

Answer

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

### Step 1: Understand the problem
We have a tower and two points A and B on the ground from which the angles of elevation to the top of the tower are given as 30° and 60°, respectively. Point A is farther away from the tower than point B.

### Step 2: Set up the diagram
Let:
- \( h \) = height of the tower
- \( a \) = distance from point A to the foot of the tower
- \( b \) = distance from point B to the foot of the tower

### Step 3: Use trigonometric ratios
From point A (angle of elevation = 30°):
\[ \tan(30°) = \frac{h}{a} \]
We know that \( \tan(30°) = \frac{1}{\sqrt{3}} \), so:
\[ \frac{1}{\sqrt{3}} = \frac{h}{a} \implies h = \frac{a}{\sqrt{3}} \quad \text{(Equation 1)} \]

From point B (angle of elevation = 60°):
\[ \tan(60°) = \frac{h}{b} \]
We know that \( \tan(60°) = \sqrt{3} \), so:
\[ \sqrt{3} = \frac{h}{b} \implies h = b\sqrt{3} \quad \text{(Equation 2)} \]

### Step 4: Equate the two expressions for height
From Equation 1 and Equation 2, we have:
\[ \frac{a}{\sqrt{3}} = b\sqrt{3} \]

### Step 5: Solve for \( a \) in terms of \( b \)
Multiplying both sides by \( \sqrt{3} \):
\[ a = 3b \]

### Step 6: Substitute back to find height
Now substitute \( a = 3b \) into either Equation 1 or Equation 2. We will use Equation 2:
\[ h = b\sqrt{3} \]
Substituting \( a \):
\[ h = \frac{3b}{\sqrt{3}} = b\sqrt{3} \]

### Step 7: Final expression for height
We can express \( h \) in terms of \( a \) or \( b \). Using \( b = \frac{a}{3} \):
\[ h = \frac{a}{3}\sqrt{3} \]

### Conclusion
The height of the tower \( h \) can be expressed as:
\[ h = \sqrt{ab} \]