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

Answer

To find the area of the rectangular field ABCD, we will use the information given about the vertical lamp post and the angles of elevation from points B and C.

### Step-by-Step Solution:

1. **Draw the Diagram:**
- Draw a rectangle ABCD with A at the bottom left, B at the bottom right, C at the top right, and D at the top left.
- Place a vertical lamp post at point A with a height of 12 m. Label the top of the lamp post as E.

2. **Identify Angles of Elevation:**
- The angle of elevation from point B to the top of the lamp post (E) is given as \(60^\circ\).
- The angle of elevation from point C to the top of the lamp post (E) is given as \(45^\circ\).

3. **Set Variables for Lengths:**
- Let the length of side AB (or CD) be \(x\) and the length of side AD (or BC) be \(y\).

4. **Use Triangle ABE:**
- In triangle ABE, we can use the tangent function:
\[ \tan(60^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{12}{x} \]
- Since \(\tan(60^\circ) = \sqrt{3}\), we have:
\[ \sqrt{3} = \frac{12}{x} \implies x = \frac{12}{\sqrt{3}} = 4\sqrt{3} \]

5. **Use Triangle ACE:**
- In triangle ACE, we can again use the tangent function:
\[ \tan(45^\circ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{12}{y} \]
- Since \(\tan(45^\circ) = 1\), we have:
\[ 1 = \frac{12}{y} \implies y = 12 \]

6. **Use Pythagorean Theorem in Triangle ADC:**
- In triangle ADC, we apply the Pythagorean theorem:
\[ AC^2 = AB^2 + AD^2 \]
- Here, \(AC\) is the hypotenuse, which is equal to the height of the lamp post, 12 m:
\[ 12^2 = x^2 + y^2 \]
- Substitute \(x = 4\sqrt{3}\) and \(y = 12\):
\[ 144 = (4\sqrt{3})^2 + 12^2 \]
\[ 144 = 48 + 144 \]
- This confirms our values are consistent.

7. **Calculate the Area of the Field:**
- The area \(A\) of the rectangle ABCD is given by:
\[ A = x \cdot y = (4\sqrt{3}) \cdot 12 = 48\sqrt{3} \]

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