A pole stands at the centre of a rectangular field and it subtends angles of `15^(@) and 45^(@)` at the mid-points of the sides of the field. If the length of its diagonal is 1200 m, then the height of the flag staff is

Options

A. 400 m
B. 200 m
C. `300sqrt(2-sqrt(3))m`
D. `300sqrt(2)-sqrt(3))m`

Answer

To find the height of the flagstaff (pole) in the rectangular field, we will follow these steps:

### Step 1: Understand the Geometry
We have a rectangular field with a diagonal of 1200 m. The pole is at the center of the field and subtends angles of 15° and 45° at the midpoints of two opposite sides of the rectangle.

### Step 2: Calculate Half of the Diagonal
The length of the diagonal (AC or BD) is given as 1200 m. The distance from the center of the rectangle to the midpoint of a side (EF) is half of the diagonal:
\[ EF = \frac{1200}{2} = 600 \text{ m} \]

### Step 3: Set Up Right Triangles
Let:
- \( O \) be the point where the pole is located (the center of the rectangle).
- \( E \) and \( F \) be the midpoints of two opposite sides of the rectangle.

From point \( O \):
- The height of the pole is \( OP = h \).
- The distance \( OE \) can be expressed using the angle at \( E \):
\[ OE = h \cdot \cot(45^\circ) = h \cdot 1 = h \]
- The distance \( OF \) can be expressed using the angle at \( F \):
\[ OF = h \cdot \cot(15^\circ) \]

### Step 4: Use Trigonometric Values
The cotangent of 15° can be calculated using the formula:
\[ \cot(15^\circ) = 2 + \sqrt{3} \]
Thus,
\[ OF = h(2 + \sqrt{3}) \]

### Step 5: Apply Pythagorean Theorem
In the right triangle \( OEF \):
\[ EF^2 = OE^2 + OF^2 \]
Substituting the values:
\[ 600^2 = h^2 + (h(2 + \sqrt{3}))^2 \]
Expanding the equation:
\[ 360000 = h^2 + h^2(2 + \sqrt{3})^2 \]

### Step 6: Expand and Simplify
Calculating \( (2 + \sqrt{3})^2 \):
\[ (2 + \sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \]
Thus, we have:
\[ 360000 = h^2 + h^2(7 + 4\sqrt{3}) \]
\[ 360000 = h^2(1 + 7 + 4\sqrt{3}) = h^2(8 + 4\sqrt{3}) \]

### Step 7: Solve for \( h^2 \)
Rearranging gives:
\[ h^2 = \frac{360000}{8 + 4\sqrt{3}} \]

### Step 8: Calculate \( h \)
Taking the square root:
\[ h = \sqrt{\frac{360000}{8 + 4\sqrt{3}}} \]
This can be simplified further.

### Step 9: Rationalize the Denominator
To simplify:
\[ h = \frac{300}{\sqrt{2 + \sqrt{3}}} \]
Rationalizing gives:
\[ h = 300 \cdot \frac{\sqrt{2 - \sqrt{3}}}{\sqrt{(2 + \sqrt{3})(2 - \sqrt{3})}} = 300 \cdot \frac{\sqrt{2 - \sqrt{3}}}{1} = 300\sqrt{2 - \sqrt{3}} \]

### Final Answer
The height of the flagstaff is:
\[ h = \frac{300}{\sqrt{2 + \sqrt{3}}} \text{ meters} \]
or equivalently,
\[ h = 300\sqrt{2 - \sqrt{3}} \text{ meters} \]