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To find the area of the triangle when two sides and the included angle are given, we can use the formula:

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

### Given values:
- \( a = 3 \) feet
- \( b = 10 \) feet
- \( C = 82^\circ 12' \)

#### Step-by-Step Solution:

1. **Convert the angle \( C = 82^\circ 12' \) into decimal form:**
   \[
   82^\circ 12' = 82 + \frac{12}{60} = 82 + 0.2 = 82.2^\circ
   \]

2. **Calculate \( \sin(82.2^\circ) \):**
   Using a calculator, \( \sin(82.2^\circ) \approx 0.9903 \).

3. **Substitute the values into the area formula:**
   \[
   \text{Area} = \frac{1}{2} \cdot 3 \cdot 10 \cdot 0.9903
   \]
   \[
   \text{Area} \approx \frac{1}{2} \cdot 3 \cdot 10 \cdot 0.9903
   \]
   \[
   \text{Area} \approx 14.8545
   \]

4. **Round to one decimal place:**
   \[
   \text{Area} \approx 14.9 \text{ square feet}
   \]

### Final Answer:
The area of the triangle is approximately **14.9 square feet**.

Would you like further clarification or additional steps explained?

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### Related Questions:
1. How do you convert degrees and minutes into decimal form?
2. Why is the sine function used in this area formula?
3. What happens if the angle is larger than 90 degrees? Does the formula change?
4. How would you calculate the area if all three sides of the triangle were known?
5. Could you find the area using a different trigonometric function?

### Tip:
When working with trigonometric functions, ensure your calculator is in degree mode if the angle is given in degrees.

Find the area of the triangle with angle C = 82°12', side a = 3 feet, and side b = 10 feet.

Learn how to find the area of a triangle when given two sides and the included angle. This example uses side lengths of 3 feet and 10 feet with an angle of 82°12', demonstrating the trigonometric area formula. Suitable for students in Grades 10-12.

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