The elevation of the top of a tower from a point on the ground is `45^(@)` . On travelling 60 m from the point towards the tower, the alevation of the top becomes `60^(@)` . The height of the tower, in metres, is

Options

A. 30
B. `30(3-sqrt(3))`
C. `30(3+ sqrt(3))`
D. `30 sqrt(3)`

Answer

To solve the problem step by step, we will use trigonometric principles and the properties of right triangles.

### Step 1: Understand the Problem
We have a tower whose height we need to find. From a point on the ground, the angle of elevation to the top of the tower is \(45^\circ\). After moving 60 meters closer to the tower, the angle of elevation becomes \(60^\circ\).

### Step 2: Draw a Diagram
Let's label the points:
- Let \(A\) be the top of the tower.
- Let \(B\) be the point where the angle of elevation is \(45^\circ\).
- Let \(C\) be the point where the angle of elevation is \(60^\circ\) after moving 60 meters towards the tower.
- Let \(h\) be the height of the tower (the vertical distance from point \(C\) to point \(A\)).
- Let \(d\) be the horizontal distance from point \(C\) to the base of the tower (point \(D\)).

### Step 3: Set Up the Equations
From point \(B\) (where the angle is \(45^\circ\)):
- The height of the tower \(h\) can be expressed as:
\[ h = d \tan(45^\circ) = d \]
(since \(\tan(45^\circ) = 1\))

From point \(C\) (where the angle is \(60^\circ\)):
- The height of the tower can also be expressed as:
\[ h = (d - 60) \tan(60^\circ) = (d - 60) \sqrt{3} \]
(since \(\tan(60^\circ) = \sqrt{3}\))

### Step 4: Equate the Two Expressions for \(h\)
Now we have two expressions for \(h\):
1. \(h = d\)
2. \(h = (d - 60) \sqrt{3}\)

Setting them equal gives:
\[ d = (d - 60) \sqrt{3} \]

### Step 5: Solve for \(d\)
Expanding the equation:
\[ d = d\sqrt{3} - 60\sqrt{3} \]
Rearranging gives:
\[ d - d\sqrt{3} = -60\sqrt{3} \]
Factoring out \(d\):
\[ d(1 - \sqrt{3}) = -60\sqrt{3} \]
Thus:
\[ d = \frac{-60\sqrt{3}}{1 - \sqrt{3}} \]

### Step 6: Rationalize the Denominator
To simplify \(d\), multiply the numerator and denominator by the conjugate of the denominator:
\[ d = \frac{-60\sqrt{3}(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} \]
Calculating the denominator:
\[ (1 - \sqrt{3})(1 + \sqrt{3}) = 1 - 3 = -2 \]
Thus:
\[ d = \frac{-60\sqrt{3}(1 + \sqrt{3})}{-2} = 30\sqrt{3}(1 + \sqrt{3}) = 30\sqrt{3} + 90 \]

### Step 7: Find \(h\)
Now substituting \(d\) back into the equation for \(h\):
\[ h = d = 30\sqrt{3} + 90 \]

### Conclusion
The height of the tower is:
\[ h = 30\sqrt{3} + 90 \text{ meters} \]