A tower is broken at a point P above the ground .The top of the tower makes an angle `60^(@)` with the ground at Q . From another point R on the opposite side of Q angle of elevation of point P is `30^(@)` .If QR =180 metre , what is the total height (in metre) of the tower ?

Options

A. 90
B. `45sqrt(3)`
C. `45(sqrt(3)-1)`
D. `45(sqrt(3)+1)`

Answer

To find the total height of the tower, we will break down the problem step by step.

### Step 1: Understand the Problem
We have a tower that is broken at point P. The top of the tower (point T) makes an angle of 60 degrees with the ground at point Q. From another point R on the opposite side of Q, the angle of elevation to point P is 30 degrees. The distance QR is given as 180 meters.

### Step 2: Set Up the Diagram
Let:
- \( h \) = height of the tower from the ground to point P.
- \( QR = 180 \) meters.
- \( RP = x \) meters (distance from R to P).
- \( PQ = 180 - x \) meters (distance from P to Q).

### Step 3: Use Trigonometric Ratios
1. **From Triangle RPQ** (where angle of elevation to P is 30 degrees):
\[ \tan(30^\circ) = \frac{h}{x} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), we have:
\[ \frac{1}{\sqrt{3}} = \frac{h}{x} \implies h = \frac{x}{\sqrt{3}} \quad \text{(Equation 1)} \]

2. **From Triangle PQT** (where angle of elevation to T is 60 degrees):
\[ \tan(60^\circ) = \frac{h}{180 - x} \]
Since \( \tan(60^\circ) = \sqrt{3} \), we have:
\[ \sqrt{3} = \frac{h}{180 - x} \implies h = \sqrt{3}(180 - x) \quad \text{(Equation 2)} \]

### Step 4: Set the Equations Equal
From Equation 1 and Equation 2, we can set them equal to each other:
\[ \frac{x}{\sqrt{3}} = \sqrt{3}(180 - x) \]
Multiply both sides by \( \sqrt{3} \):
\[ x = 3(180 - x) \]
Expanding gives:
\[ x = 540 - 3x \]
Combining like terms:
\[ 4x = 540 \implies x = 135 \text{ meters} \]

### Step 5: Find the Height \( h \)
Now substitute \( x = 135 \) back into Equation 1:
\[ h = \frac{135}{\sqrt{3}} = 45\sqrt{3} \text{ meters} \]

### Step 6: Total Height of the Tower
The total height of the tower \( H \) is:
\[ H = h + PQ = h + (180 - x) = 45\sqrt{3} + (180 - 135) = 45\sqrt{3} + 45 \]

### Final Calculation
Using \( \sqrt{3} \approx 1.732 \):
\[ H \approx 45 \times 1.732 + 45 \approx 77.94 + 45 \approx 122.94 \text{ meters} \]
Thus, the total height of the tower is approximately \( 122.94 \) meters.

### Summary
The total height of the tower is \( 45\sqrt{3} \) meters or approximately \( 122.94 \) meters.