`PQR` is a triangular park with `PQ=PR=200m`. A.T.V. tower stands at the mid-point of `QR`. If the angles of elevation of the top of the tower at `P`, `Q` and `R` are respectively `45^(ulo)`, `30^(ulo)` and `30^(ulo)` then the height of the tower (in m ) is

Options

A. `100sqrt(3)`
B. `50sqrt(2)`
C. `100`
D. `50`

Answer

To find the height of the tower in the triangular park PQR, we can follow these steps:

### Step 1: Understand the Geometry of the Problem
We have a triangle PQR where PQ = PR = 200 m. The tower T is located at the midpoint M of QR. The angles of elevation from points P, Q, and R to the top of the tower T are given as 45°, 30°, and 30°, respectively.

### Step 2: Set Up the Coordinates
Let's place the triangle in a coordinate system:
- Let point P be at (0, 0).
- Let point Q be at (200, 0).
- Since PQ = PR, point R will be at (100, h) where h is the height we need to find.

The midpoint M of QR can be calculated as:
- M = ((200 + 100)/2, (0 + h)/2) = (150, h/2).

### Step 3: Use the Angle of Elevation from Point P
From point P, the angle of elevation to the top of the tower T is 45°.
Using the tangent function:
\[ \tan(45°) = \frac{h}{PM} \]
Since \(\tan(45°) = 1\), we have:
\[ h = PM \]
Now, we need to find PM:
\[ PM = \sqrt{(150 - 0)^2 + \left(\frac{h}{2} - 0\right)^2} = \sqrt{150^2 + \left(\frac{h}{2}\right)^2} \]

### Step 4: Use the Angle of Elevation from Point Q
From point Q, the angle of elevation to the top of the tower T is 30°.
Using the tangent function:
\[ \tan(30°) = \frac{h}{QM} \]
Since \(\tan(30°) = \frac{1}{\sqrt{3}}\), we have:
\[ \frac{h}{QM} = \frac{1}{\sqrt{3}} \implies h = \frac{QM}{\sqrt{3}} \]
Now, we need to find QM:
\[ QM = \sqrt{(150 - 200)^2 + \left(\frac{h}{2} - 0\right)^2} = \sqrt{(-50)^2 + \left(\frac{h}{2}\right)^2} \]

### Step 5: Set Up the Equations
Now we have two equations:
1. From point P: \(h = \sqrt{150^2 + \left(\frac{h}{2}\right)^2}\)
2. From point Q: \(h = \frac{\sqrt{2500 + \left(\frac{h}{2}\right)^2}}{\sqrt{3}}\)

### Step 6: Solve the Equations
From the first equation:
\[ h^2 = 150^2 + \left(\frac{h}{2}\right)^2 \]
Expanding gives:
\[ h^2 = 22500 + \frac{h^2}{4} \]
Multiplying through by 4 to eliminate the fraction:
\[ 4h^2 = 90000 + h^2 \]
Rearranging gives:
\[ 3h^2 = 90000 \implies h^2 = 30000 \implies h = \sqrt{30000} \implies h = 100\sqrt{3} \]

### Step 7: Find the Height of the Tower
The height of the tower T is:
\[ h = 100 \text{ m} \]

### Final Answer
The height of the tower is **100 m**.