To find the length of side \(a\) in triangle \(ABC\) given the area and the lengths of side \(b\) and the angle \(C\), we can use the formula for the area of a triangle:
\[ \text{Area} = \frac{1}{2} \cdot b \cdot a \cdot \sin(C) \]
We know:
- The area of the triangle \(\text{Area} = 134\) square feet,
- \(b = 24\) feet,
- \(C = 41^\circ\).
Substituting the known values into the formula gives:
\[ 134 = \frac{1}{2} \cdot 24 \cdot a \cdot \sin(41^\circ) \]
Now, we can solve for \(a\):
- First, calculate \(\sin(41^\circ)\):
\[ \sin(41^\circ) \approx 0.6561 \]
- Substitute this value into the equation:
\[ 134 = \frac{1}{2} \cdot 24 \cdot a \cdot 0.6561 \]
- Simplify:
\[ 134 = 12 \cdot a \cdot 0.6561 \]
- Further simplification:
\[ 134 = 7.8732 \cdot a \]
- Now, solve for \(a\):
\[ a = \frac{134}{7.8732} \approx 17.0 \]
Rounding \(17.0\) to the nearest whole number gives:
\[ a \approx 17 \text{ feet} \]
Thus, the length of side \(a\) is 17 feet.
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