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A goat farmer builds a triangular shape enclosure for his new herd of goats. The enclosure is shown below. He needs to know the area of the enclosure so that he can get an appropriate number of goats for the size of the enclosure. What is the area of the enclosure? Round to the nearest tenth of a square yard. 


To find the area of the triangular enclosure, we can use the formula for the area of a triangle based on two sides and the included angle, which we derived earlier:

\[
A = \frac{1}{2} ab \sin(\theta)
\]

### Given:
- Side \( a = 250 \) yards
- Side \( b = 525 \) yards
- Angle \( \theta = 33^\circ \)

### Step 1: Plug in the values into the area formula.
\[
A = \frac{1}{2} \times 250 \times 525 \times \sin(33^\circ)
\]

### Step 2: Calculate the sine of \( 33^\circ \).
\[
\sin(33^\circ) \approx 0.5446
\]

### Step 3: Calculate the area.
\[
A = \frac{1}{2} \times 250 \times 525 \times 0.5446
\]

\[
A \approx \frac{1}{2} \times 250 \times 285.915
\]

\[
A \approx \frac{1}{2} \times 142957.5
\]

\[
A \approx 71478.75 \text{ square yards}
\]

### Step 4: Round to the nearest tenth.
\[
A \approx 71478.8 \text{ square yards}
\]

### Final Answer:
The area of the triangular enclosure is approximately \( \boxed{71478.8} \) square yards.

Would you like further details, or do you have any questions?

Here are some related questions you might find interesting:
1. How would the area change if the angle were different?
2. Can you find the length of the third side of the triangle using the law of cosines?
3. How can you verify the result using another method, like Heron's formula?
4. What if the angle was measured in radians? How would that affect the calculation?
5. How does the accuracy of the sine value impact the area calculation?

**Tip:** When rounding off in trigonometric calculations, be consistent with the level of precision to maintain accuracy in the final result.

Discover how to calculate the area of a triangular enclosure with sides of 250 yards and 525 yards, and an included angle of 33 degrees. This problem involves trigonometric principles and the formula for the area of a triangle. Suitable for high school students.

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