ABCD is a rectangular field. A vertical lamp post of height 12 m stands at the corner A. If the angle of elevation of its top from B is `60^(@)` and from C is `45^(@)`, then the area of the field is

Options

A. `48sqrt(2)` sq .m
B. `48sqrt(3)` sq.m
C. `48` sq m
D. `12sqrt(2)` sq. m

Answer

To find the area of the rectangular field ABCD, we need to determine the lengths of sides AB and AC using the given angles of elevation from points B and C to the top of the lamp post at point A.

### Step 1: Identify the given information
- Height of the lamp post (h) = 12 m
- Angle of elevation from point B = 60°
- Angle of elevation from point C = 45°

### Step 2: Use trigonometry to find distances
We will use the tangent function, which relates the angle of elevation to the opposite side (height of the lamp post) and the adjacent side (distance from the point to the base of the lamp post).

**From point B:**
- Let the distance from B to A be \( x \).
- Using the tangent of the angle of elevation:
\[ \tan(60°) = \frac{\text{height}}{\text{distance}} = \frac{12}{x} \]
\[ \sqrt{3} = \frac{12}{x} \implies x = \frac{12}{\sqrt{3}} = 4\sqrt{3} \text{ m} \]

**From point C:**
- Let the distance from C to A be \( y \).
- Using the tangent of the angle of elevation:
\[ \tan(45°) = \frac{12}{y} \]
\[ 1 = \frac{12}{y} \implies y = 12 \text{ m} \]

### Step 3: Determine the dimensions of the rectangle
- The length of side AB (x) = \( 4\sqrt{3} \) m
- The length of side AC (y) = 12 m

### Step 4: Calculate the area of the rectangle
The area \( A \) of rectangle ABCD is given by:
\[ A = \text{length} \times \text{breadth} = AB \times AC = (4\sqrt{3}) \times 12 \]
\[ A = 48\sqrt{3} \text{ m}^2 \]

### Final Answer
The area of the rectangular field ABCD is \( 48\sqrt{3} \text{ m}^2 \).

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