If the sides 50 m and 130 m of the triangular field meet at an angle of `72^(@)` then find the area in which wheat is cultivated `(Sin 72^(@) = 0.9510, Cos 72^(@) = 0.309)`.

Options

A. `100 pi m^(2)`
B. `125 pi m^(2)`
C. `160 pi m^(2)`
D. None of these

Answer

To find the area of the triangular field where wheat is cultivated, we can use the formula for the area of a triangle when two sides and the included angle are known. The formula is:

\[ \text{Area} = \frac{1}{2} \times a \times b \times \sin(C) \]

Where:
- \( a \) and \( b \) are the lengths of the two sides,
- \( C \) is the angle between those sides.

### Step 1: Identify the values
Given:
- Side \( a = 50 \, m \)
- Side \( b = 130 \, m \)
- Angle \( C = 72^\circ \)
- \( \sin(72^\circ) = 0.9510 \)

### Step 2: Substitute the values into the formula
Substituting the given values into the area formula:

\[ \text{Area} = \frac{1}{2} \times 50 \times 130 \times \sin(72^\circ) \]

### Step 3: Calculate the area
Now, we can calculate the area:

\[ \text{Area} = \frac{1}{2} \times 50 \times 130 \times 0.9510 \]

Calculating \( 50 \times 130 \):

\[ 50 \times 130 = 6500 \]

Now, substituting this back into the area formula:

\[ \text{Area} = \frac{1}{2} \times 6500 \times 0.9510 \]

Calculating \( \frac{1}{2} \times 6500 \):

\[ \frac{1}{2} \times 6500 = 3250 \]

Now, multiplying by \( 0.9510 \):

\[ \text{Area} = 3250 \times 0.9510 = 3095.75 \, m^2 \]

### Step 4: Final area
Thus, the area in which wheat is cultivated is approximately:

\[ \text{Area} \approx 3095.75 \, m^2 \]